7.13 - Inequalities

7.13  The student will solve one- and two-step 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

2016 VDOE Curriculum Framework - 7.13 Understanding

·  A one-step inequality may include, but not be limited to, inequalities such as the following: 2x > 5; y - 3 £ −6;  x < -3;  a − (4) ≥ 11.

·  A two-step 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 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 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

·  7.131  Apply properties of real numbers and the multiplication and division properties of inequality to solve one-step inequalities in one variable, and the addition, subtraction, multiplication, and division properties of inequality to solve two-step inequalities in one variable. Coefficients and numeric terms will be rational.

·  7.135  Solve practical problems that require the solution of a one- or two-step inequality.

·  7.136  Identify a numerical value(s) that is part of the solution set of a given inequality.

·  7.132  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

Updated: Nov 20, 2018