Compound Inequalities: Solutions & Non-Solutions

Contribute email twitter instagram facebook

Lessons

Standards in this Lesson

Common Core Math Standards

6.EE.B: Reason about and solve one-variable equations and inequalities.
7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
HSF.BF.A: Build a function that models a relationship between two quantities.
HSF.BF.A.1: Write a function that describes a relationship between two quantities.
HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
HSF.IF.B: Interpret functions that arise in applications in terms of the context.
HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

CSTA Standards

3B-AP-10: Use and adapt classic algorithms to solve computational problems.

Alabama Standards

AL.6.EE.B: Reason about and solve one-variable equations and inequalities.
AL.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
AL.HSF.BF.A: Build a function that models a relationship between two quantities.
AL.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
AL.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
AL.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
AL.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

Arkansas Standards

AR.6.EE.B: Reason about and solve one-variable equations and inequalities.
AR.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
AR.HSF.BF.A: Build a function that models a relationship between two quantities.
AR.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
AR.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
AR.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
AR.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

California Standards

CA.6.EE.B: Reason about and solve one-variable equations and inequalities.
CA.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
CA.HSF.BF.A: Build a function that models a relationship between two quantities.
CA.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
CA.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
CA.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
CA.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

Colorado Standards

CO.6.EE.B: Reason about and solve one-variable equations and inequalities.
CO.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
CO.HSF.BF.A: Build a function that models a relationship between two quantities.
CO.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
CO.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
CO.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
CO.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

Connecticut Standards

CT.6.EE.B: Reason about and solve one-variable equations and inequalities.
CT.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
CT.HSF.BF.A: Build a function that models a relationship between two quantities.
CT.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
CT.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
CT.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
CT.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

Delaware Standards

DE.6.EE.B: Reason about and solve one-variable equations and inequalities.
DE.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
DE.HSF.BF.A: Build a function that models a relationship between two quantities.
DE.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
DE.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
DE.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
DE.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

Delaware Standards

GA.6.EE.B: Reason about and solve one-variable equations and inequalities.
GA.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
GA.HSF.BF.A: Build a function that models a relationship between two quantities.
GA.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
GA.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
GA.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
GA.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

Hawaii Standards

HI.6.EE.B: Reason about and solve one-variable equations and inequalities.
HI.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
HI.HSF.BF.A: Build a function that models a relationship between two quantities.
HI.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
HI.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
HI.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
HI.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

Idaho Standards

ID.6.EE.B: Reason about and solve one-variable equations and inequalities.
ID.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
ID.HSF.BF.A: Build a function that models a relationship between two quantities.
ID.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
ID.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
ID.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
ID.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

Illinois Standards

IL.6.EE.B: Reason about and solve one-variable equations and inequalities.
IL.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
IL.HSF.BF.A: Build a function that models a relationship between two quantities.
IL.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
IL.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
IL.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
IL.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

Iowa Standards

IA.6.EE.B: Reason about and solve one-variable equations and inequalities.
IA.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
IA.HSF.BF.A: Build a function that models a relationship between two quantities.
IA.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
IA.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
IA.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
IA.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

Kansas Standards

KS.6.EE.B: Reason about and solve one-variable equations and inequalities.
KS.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
KS.HSF.BF.A: Build a function that models a relationship between two quantities.
KS.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
KS.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
KS.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
KS.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

Kentucky Standards

KY.6.EE.B: Reason about and solve one-variable equations and inequalities.
KY.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
KY.HSF.BF.A: Build a function that models a relationship between two quantities.
KY.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
KY.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
KY.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
KY.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

Louisiana Standards

LA.6.EE.B: Reason about and solve one-variable equations and inequalities.
LA.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
LA.HSF.BF.A: Build a function that models a relationship between two quantities.
LA.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
LA.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
LA.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
LA.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

Maine Standards

ME.6.EE.B: Reason about and solve one-variable equations and inequalities.
ME.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
ME.HSF.BF.A: Build a function that models a relationship between two quantities.
ME.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
ME.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
ME.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
ME.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

Michigan Standards

MI.6.EE.B: Reason about and solve one-variable equations and inequalities.
MI.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
MI.HSF.BF.A: Build a function that models a relationship between two quantities.
MI.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
MI.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
MI.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
MI.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

Maryland Standards

MD.6.EE.B: Reason about and solve one-variable equations and inequalities.
MD.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
MD.HSF.BF.A: Build a function that models a relationship between two quantities.
MD.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
MD.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
MD.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
MD.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

Massachusetts Standards

MA.6-8.CT.a.2: Define a simple function that represents a more complex task/problem and can be reused to solve similar tasks/problems.
MA.6-8.CT.b.2: Use logical reasoning to predict outputs given varying inputs.
MA.6-8.CT.d.2: Use functions to hide the detail in a program.
MA.6-8.CT.d.5: Trace programs step-by-step in order to predict their behavior.
MA.6.EE.B: Reason about and solve one-variable equations and inequalities.
MA.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
MA.9-12.CT.b.2: Represent algorithms using structured language, such as pseudocode.
MA.HSF.BF.A: Build a function that models a relationship between two quantities.
MA.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
MA.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
MA.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
MA.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

Mississippi Standards

MS.6.EE.B: Reason about and solve one-variable equations and inequalities.
MS.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
MS.HSF.BF.A: Build a function that models a relationship between two quantities.
MS.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
MS.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
MS.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
MS.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

Missouri Standards

MO.6.EE.B: Reason about and solve one-variable equations and inequalities.
MO.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
MO.HSF.BF.A: Build a function that models a relationship between two quantities.
MO.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
MO.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
MO.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
MO.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

Montana Standards

MT.6.EE.B: Reason about and solve one-variable equations and inequalities.
MT.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
MT.HSF.BF.A: Build a function that models a relationship between two quantities.
MT.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
MT.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
MT.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
MT.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

New Hampshire Standards

NH.6.EE.B: Reason about and solve one-variable equations and inequalities.
NH.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
NH.HSF.BF.A: Build a function that models a relationship between two quantities.
NH.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
NH.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
NH.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
NH.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

Nevada Standards

NV.6.EE.B: Reason about and solve one-variable equations and inequalities.
NV.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
NV.HSF.BF.A: Build a function that models a relationship between two quantities.
NV.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
NV.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
NV.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
NV.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

New Jersey Standards

NJ.6.EE.B: Reason about and solve one-variable equations and inequalities.
NJ.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
NJ.HSF.BF.A: Build a function that models a relationship between two quantities.
NJ.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
NJ.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
NJ.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
NJ.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

New Mexico Standards

NM.6.EE.B: Reason about and solve one-variable equations and inequalities.
NM.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
NM.HSF.BF.A: Build a function that models a relationship between two quantities.
NM.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
NM.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
NM.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
NM.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

North Carolina Standards

NC.6.EE.B: Reason about and solve one-variable equations and inequalities.
NC.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
NC.HSF.BF.A: Build a function that models a relationship between two quantities.
NC.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
NC.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
NC.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
NC.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

North Dakota Standards

ND.6.EE.B: Reason about and solve one-variable equations and inequalities.
ND.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
ND.HSF.BF.A: Build a function that models a relationship between two quantities.
ND.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
ND.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
ND.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
ND.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

New York Standards

NY.6.EE.B: Reason about and solve one-variable equations and inequalities.
NY.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
NY.HSF.BF.A: Build a function that models a relationship between two quantities.
NY.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
NY.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
NY.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
NY.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

Oklahoma Standards

OK.6.A.3.1: Represent real-world or mathematical situations using expressions, equations and inequalities involving variables and rational numbers.
OK.7.A.3.3: Represent real-world or mathematical situations using equations and inequalities involving variables and rational numbers.
OK.A1.A.2.2: Represent relationships in various contexts with compound and absolute value inequalities and solve the resulting inequalities by graphing and interpreting the solutions on a number line.
OK.A1.D.2.2: Describe the concepts of intersections, unions, and complements using Venn diagrams to evaluate probabilities. Understand the relationships between these concepts and the words AND, OR, and NOT.
OK.PA.A.4.3: Represent real-world situations using equations and inequalities involving one variable.

Ohio Standards

OH.6.EE.B: Reason about and solve one-variable equations and inequalities.
OH.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
OH.HSF.BF.A: Build a function that models a relationship between two quantities.
OH.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
OH.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
OH.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
OH.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

Oregon Standards

OR.6.EE.B: Reason about and solve one-variable equations and inequalities.
OR.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
OR.HSF.BF.A: Build a function that models a relationship between two quantities.
OR.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
OR.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
OR.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
OR.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

Pennsylvania Standards

PA.6.EE.B: Reason about and solve one-variable equations and inequalities.
PA.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
PA.HSF.BF.A: Build a function that models a relationship between two quantities.
PA.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
PA.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
PA.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
PA.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

Rhode Island Standards

RI.6.EE.B: Reason about and solve one-variable equations and inequalities.
RI.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
RI.HSF.BF.A: Build a function that models a relationship between two quantities.
RI.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
RI.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
RI.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
RI.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

South Dakota Standards

SD.6.EE.B: Reason about and solve one-variable equations and inequalities.
SD.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
SD.HSF.BF.A: Build a function that models a relationship between two quantities.
SD.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
SD.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
SD.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
SD.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

Tennessee Standards

TN.6.EE.B: Reason about and solve one-variable equations and inequalities.
TN.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
TN.HSF.BF.A: Build a function that models a relationship between two quantities.
TN.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
TN.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
TN.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
TN.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

Utah Standards

UT.6.EE.B: Reason about and solve one-variable equations and inequalities.
UT.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
UT.HSF.BF.A: Build a function that models a relationship between two quantities.
UT.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
UT.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
UT.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
UT.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

Vermont Standards

VT.6.EE.B: Reason about and solve one-variable equations and inequalities.
VT.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
VT.HSF.BF.A: Build a function that models a relationship between two quantities.
VT.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
VT.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
VT.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
VT.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

Washington Standards

WA.6.EE.B: Reason about and solve one-variable equations and inequalities.
WA.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
WA.HSF.BF.A: Build a function that models a relationship between two quantities.
WA.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
WA.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
WA.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
WA.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

West Virginia Standards

WV.6.EE.B: Reason about and solve one-variable equations and inequalities.
WV.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
WV.HSF.BF.A: Build a function that models a relationship between two quantities.
WV.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
WV.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
WV.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
WV.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

Wisconsin Standards

WI.6.EE.B: Reason about and solve one-variable equations and inequalities.
WI.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
WI.HSF.BF.A: Build a function that models a relationship between two quantities.
WI.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
WI.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
WI.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
WI.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

Wyoming Standards

WY.6.EE.B: Reason about and solve one-variable equations and inequalities.
WY.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
WY.HSF.BF.A: Build a function that models a relationship between two quantities.
WY.HSF.BF.A.1: Write a function that describes a relationship between two quantities.
WY.HSF.IF.A.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
WY.HSF.IF.B: Interpret functions that arise in applications in terms of the context.
WY.HSS.CP.A.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ('or', 'and', 'not').

Textbook Alignment

IM Algebra 1

IM.Alg1.2.18: Representing Situations with Inequalities

Practices in this Lesson

Math

MP.1: Make sense of problems and persevere in solving them

Math Lang. Routines

MLR.5: Co-Craft Questions and Problems

(Also available in Pyret)

Students build upon their understanding of Booleans and simple inequalities to compose compound inequalities using the concepts of union and intersection.

Lesson Goals

Students will be able to:

Understand how the conjunctions and and or differ
Describe how functions can work together
Describe the solution set of a compound inequality

Student-Facing Lesson Goals

I can explain the difference between how and and or are used in programming
I can use two or more inequalities together and describe the area they enclose

Materials

Glossary

compound inequality: an inequality that combines two simple inequalities using and or or
function: a relation from a set of inputs to a set of possible outputs, where each input is related to exactly one output
intersection: the set of values that makes both inequalities true
union: the set of values that makes either or both of a set of inequalities true

🔗Introducing Compound Inequalities

Overview

Students consider the need to compose inequalities, and think about how to write them.

Launch

We use inequalities for lots of things:

Is it hot out? ( $\displaystyle temperature \gt 80°$ )
Did I get paid enough for painting that fence? ( $\displaystyle paid \ge $100$ )
Are the cookies finished baking? ( $\displaystyle timer = 0$ )

What other examples can you come up with?

Many times we need to combine inequalities:

Should I go to the beach? ( $\displaystyle temperature \gt 80°$ and $\displaystyle weather = "sunny"$ )
Was this burrito worth the price? ( $\displaystyle taste = "delicious"$ and $\displaystyle price \leq $15$ )

Can you think of examples of when we might want to combine inequalities?

Guide students through other examples of and and or with various statements, such as:

"I’m wearing a red shirt AND I’m a math teacher, true or false?"
"I’m an NBA basketball star OR I’m having pizza for lunch, true or false?"

This can make for a good sit-down, stand-up activity, where students take turns saying compound Boolean statements and everyone stands if that statement is true for them.

Investigate

Both mathematics and programming have ways of combining - or composing - inequalities.

Complete Converting Circles of Evaluation to Code and Compound Inequalities — Practice.

Synthesize

Be really careful to check for understanding here.

Expressions using and only produce true if both of their sub-expressions are true.
Expressions using or produce true if either of their sub-expressions are true.

Strategies for English Language Learners

When describing compound inequalities, be careful not to use "English shortcuts". For example, we might say "I am holding a marker and an eraser" instead of "I am holding a marker and I am holding an eraser." These sentences mean the same thing, but the first one obscures the fact that "and" joins two complete phrases. For ELL/ESL students, this is unecessarily adds to cognitive load!

🔗Solutions and Non-Solutions of Compound Inequalities

Launch

The questions below could be printed from Compound Inequality Warmup.

What are 4 solutions for $\displaystyle x \gt 5$ ?
- Possible answers: 5.001, 1000, 24, 73.26
What are 4 non-solutions for $\displaystyle x \gt 5$ ?
- Possible answers: 5, -1000, 4.9, 0
What are 4 solutions for $\displaystyle x \le 15$ ?
- Possible answers: 15, 14.98, -63, 1/2
What are 4 non-solutions for $\displaystyle x \le 15$ ?
- Possible answers: 15.1, 25, 2022, 1000000
What numbers are in the solutions set of $\displaystyle x \gt 5$ and $\displaystyle x \le 15$ , making both of these inequalities true?
- Possible answers: 5.001, 10, 12.374, 15
How would that be different from the solution set of $\displaystyle x \gt 5$ or $\displaystyle x \le 15$ , making at least one of these inequalities true?
- All real numbers are a part of the solution set for $\displaystyle x \gt 5$ or $\displaystyle x \le 15$ because there aren’t any numbers that don’t satisfy at least one of these expressions.

Investigate

Open the Compound Inequalities Starter File.
Click "Run" to see the graphs you’ve just considered.

When students click "Run", four graphs will appear. The top two are the simple inequalities they’ve just discussed. Encourage students to verify that their solutions and non-solutions are correct. Explain that the bottom two graphs are produced by the special functions and-intersection and or-union. Use the discussion questions below as a jumping off point in guiding students' analysis of the starter file. These questions could also be printed from Exploring Compound Inequality.

What does and-intersection do?
- It takes in two functions and a list of numbers and produces a graph with the points and the shaded intersection of values that make both of the inequalities true.

A compound inequality displayed on a number line, showing that the inequalities intersect between 5 and 15. The dot on 5 is red. The dot on 15 is green. 🖼Show image

Why is the circle on 5 red and the circle on 15 green?
- Because 5 is not part of the solution - it’s not bigger than itself. And 15 is part of the solution - it’s less than or equal to 15.
Do you think every graph made with and-intersection will have different color dots at the ends? Why or why not?
- No. It will depend on whether or not the inequality symbols include an equal sign. Sometimes one will and one won’t. Sometimes neither will. Sometimes both will.
What does or-union do?
- It takes in two functions and a list of numbers and produces a graph with the points and the shaded union of values that make either or both of the inequalities true.

A compound inequality displayed on a number line, showing an infinite union 🖼Show image

Why did the graph of this or-union result in the whole number line being shaded blue?
- Because in order to make an or statement true, a number only has to make one of the inequalities true. Every number in the universe is either greater than 5 or less than or equal to 15. There aren’t any non-solutions!
Not all graphs of or-union will look like this. Can you think of a pair of inequalities whose union won’t shade the whole graph?
- Answers will vary! $\displaystyle x \lt 1$ or $\displaystyle x \lt 3$

Change the function definition on line 8 to $\displaystyle x \lt 5$ and the definition on line 9 to $\displaystyle x \ge 15$ and, before you click "Run", take a moment to think about what the new graphs of and-intersection and or-union will look like. Then click "Run" and take a look.

What does the new and-intersection graph look like?
- We see a graph with only red circles, representing non-solutions, as below. None of the graph will be shaded blue, because there aren’t any numbers that are both smaller than 5 and greater than or equal to 15. A message will tell us that no solutions exist in the provided range. 🖼Show image
What does the new or-union graph look like?
- There’s an arrow to the left and an arrow to the right with a break in the middle between 5 and 15! The dot for 5 is red because it’s not part of the solution. The dot for 15 is green because it’s part of the solution. 🖼Show image
Why is the dot for 5 red and the dot for 15 green?
- The dot for 5 is red because it’s not part of the solution. The dot for 15 is green because it’s part of the solution.
Which of the 8 numbers from the list are part of the solution set? How do you know?
- -5, -2.1, 0, 15, 20… the circles are green.
Is 3 part of the solution set? How do you know?
- Yes. It’s in the blue shaded region.
Is 10 part of the solution set? How do you know?
- No. It’s in the unshaded area.

Once students are familiar with the starter file, they are ready to use it as they practice identifying solutions and non-solutions for compound inequalities.

Turn to Compound Inequalities: Solutions & Non-Solutions.
Explore the compound inequalities listed using the Compound Inequalities Starter File, identifying solutions and non-solutions for each.

Explain to students that instead of defining two functions as simple inequalities, we can produce an inequality graph by defining one function to be a compound inequality! In the following activity, students will analyze inequality graphs and define a single function that produces the graph. Walk students through the completed first example before they attempt to write this code on their own.

Turn to Compound Inequality Functions.
Write code to describe the compound inequalities pictured.

Synthesize

How did the graphs of intersections and unions differ?

🔗Additional Exercises:

Converting Circles of Evaluation with Booleans to Code 2

These materials were developed partly through support of the National Science Foundation, (awards 1042210, 1535276, 1648684, and 1738598). Bootstrap by the Bootstrap Community is licensed under a Creative Commons 4.0 Unported License. This license does not grant permission to run training or professional development. Offering training or professional development with materials substantially derived from Bootstrap must be approved in writing by a Bootstrap Director. Permissions beyond the scope of this license, such as to run training, may be available by contacting contact@BootstrapWorld.org.