Math  201920
7.13  Inequalities
7.13 The student will solve one and twostep linear inequalities in one variable, including practical problems, involving addition, subtraction, multiplication, and division, and graph the solution on a number line.
BIG IDEAS
 I can avoid receiving a speeding ticket when I can drive and
won’t go over the allowed number of text messages per
month on my cell phone.
 I will be
able to compare two mathematical statements that are not equal.
UNDERSTANDING THE STANDARD
· A onestep inequality may include, but not be limited to, inequalities such as the following: 2x > 5; y  3 £ −6; x < 3; a − (−4) ≥ 11.
·
A twostep
inequality may include, but not be limited to inequalities such as the
following:
2x + 1 < −25; 2x +
≥ 5; −25 > 7.2x + 1; £ 4; x – 2 £ 10.
· The solution set to an inequality is the set of all numbers that make the inequality true.
· The inverse operation for addition is subtraction, and the inverse operation for multiplication is division.
· The procedures for solving inequalities are the same as those to solve equations except for the case when an inequality is multiplied or divided on both sides by a negative number. Then the inequality sign is changed from less than to greater than, or greater than to less than.
· When both expressions of an inequality are multiplied or divided by a negative number, the inequality symbol reverses (e.g., –3x < 15 is equivalent to x > –5).
· Solutions to inequalities can be represented using a number line.
· In an inequality, there can be more than one value for the variable that makes the inequality true. There can be many solutions. (i.e., x + 4 > −3 then the solution is x > −7. This means that x can be any number greater than 7. A few solutions might be 6.5, 3, 0, 4, 25, etc.)
· Properties of real numbers and properties of inequality can be used to solve inequalities, justify solutions, and express simplification. Students should use the following properties, where appropriate, to further develop flexibility and fluency in problem solving (limitations may exist for the values of a, b, or c in this standard).
 Commutative property of addition: .
 Commutative property of multiplication: .
 Subtraction and division are not commutative.
 The additive identity is zero (0) because any number added to zero is the number. The multiplicative identity is one (1) because any number multiplied by one is the number. There are no identity elements for subtraction and division.
 Identity property of addition (additive identity property): .
 Identity property of multiplication (multiplicative identity property): .

Inverses are
numbers that combine with other numbers and result in identity elements
(e.g., 5 + (–5) = 0; · 5 = 1).
 Inverse property of addition (additive inverse property): .
 Inverse property of multiplication (multiplicative inverse property): .
 Zero has no multiplicative inverse.
 Multiplicative property of zero: .
 Division by zero is not a possible mathematical operation. It is undefined.
 Substitution property: If , then b can be substituted for a in any expression, equation, or inequality.
 Addition property of inequality: If then; if then .
 Subtraction property of inequality: If then; if then ..

Multiplication
property of inequality: If then ; if
then .

Multiplication
property of inequality (multiplication by a negative number): If
then ; if then .
 Division property of inequality: If then ; if then .
 Division property of inequality (division by a negative number): If then ; if then .
ESSENTIALS
 How are the
procedures for solving equations and inequalities the same?
The procedures are the same except for the case when an inequality is multiplied or divided on both sides by a negative number. Then the inequality sign is changed from less than to greater than, or greater than to less than.  How is the
solution to an inequality different from that of a linear equation?
In an inequality, there can be more than one value for the variable that makes the inequality true.
The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to
·
7.13_{1} Apply
properties of real numbers and the multiplication and division properties of
inequality to solve onestep
inequalities in one variable, and the addition, subtraction, multiplication,
and division properties of inequality to solve
twostep inequalities in one variable. Coefficients and numeric terms will be
rational.
· 7.13_{5} Solve practical problems that require the solution of a one or twostep inequality.
· 7.13_{6} Identify a numerical value(s) that is part of the solution set of a given inequality.
· 7.13_{2} Represent solutions to inequalities algebraically and graphically using a number line.
KEY VOCABULARY
inequalities, variable, solutions, number line, operation, inverse, inverse operations, algebraic sentences, < , >, <, >, identify, graph, demonstrate