# 6.14 - Inequalities

6.14  The student will

a)  represent a practical situation with a linear inequality in one variable; and

b)  solve one-step linear inequalities in one variable, involving addition or subtraction, 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 - 6.14 Understanding

·  The solution set to an inequality is the set of all numbers that make the inequality true.

·  Inequalities can represent practical situations.

Example: Jaxon works at least 4 hours per week mowing lawns. Write an inequality representing this situation and graph the solution.

x ≥ 4 or 4 £ x

Students might then be asked: “Would Jaxon ever work 3 hours in a week? 6 hours?

·  The variable in an inequality may represent values that are limited by the context of the problem or situation. Example: if the variable represents all children in a classroom who are taller than 36 inches, the variable will be limited to have a minimum and maximum value based on the heights of the children. Students are not expected to represent these situations with a compound inequality (e.g., 36 < x < 70) but only recognize that the values satisfying the single inequality (x > 36) will be limited by the context of the situation.

·  Inequalities using the < or > symbols are represented on a number line with an open circle on the number and a shaded line over the solution set.

Example: When graphing x < 4, use an open circle above the 4 to indicate that the 4 is not included.

·  Inequalities using the £ or ≥ symbols are represented on a number line with a closed circle on the number and shaded line in the direction of the solution set.

Example: When graphing x  4 fill in the circle above the 4 to indicate that the 4 is included.

·  It is important for students to see inequalities written with the variable before the inequality symbol and after.  Example: x >5 is not the same relationship as 5 > x. However, x > 5 is the same relationship as 5 < x.

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

·  Solving an equation or inequality involves a process of determining which value(s) from a specified set, if any, make the equation or inequality a true statement. Substitution can be used to determine whether a given value(s) makes an equation or inequality true.

·  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 neither commutative nor associative.

-  Identity property of multiplication (multiplicative identity property):

-  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.

-  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:

-  Addition property of inequality: If  then; if  then  (this property also applies to .

-  Subtraction property of inequality: If  then; if  then  (this property also applies to .

-  Substitution property: If  then b can be substituted for a in any expression, equation or inequality.

### ESSENTIALS

• In an inequality, does the order of the elements matter?
Yes, the order does matter. For example, x > 5 is not  the same relationship as 5 > x. However, x > 5 is the same relationship as 5 < x.

The student will use problem solving, mathematical communication, mathematical reasoning, connections and representation to

·  6.14a1  Given a verbal description, represent a practical situation with a one-variable linear inequality.

·  6.14b1  Apply properties of real numbers and the addition or subtraction property of inequality to solve a one-step linear inequality in one variable, and graph the solution on a number line.  Numeric terms being added or subtracted from the variable are limited to integers.

·  6.14b2  Given the graph of a linear inequality with integers, represent the inequality two different ways (e.g., x < -5 or -5 > x) using symbols.

·  6.14ab1  Identify a numerical value(s) that is part of the solution set of a given inequality.

### KEY VOCABULARY

inequality, number line, open circle, closed circle, variable, integer, solution set

Updated: Nov 20, 2018