SECTION 27 SOLVING LINEAR INEQUALITIES LINEAR INEQUALITY IN ONE

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Section 2

Section 2.7 Solving Linear Inequalities

Linear Inequality in One Variable:

A linear inequality in one variable is an inequality that can be written in the form ax + b < c where a, b, and c are real numbers and a ≠ 0. This also holds true for the inequality symbols >, ≤, and ≥.

Words	Less Than	Less Than or Equal To	Greater Than	Greater Than or Equal To
Inequality Sign	<	≤	>	≥
Number Line Dot	Open dot ○	Closed dot ●	Open dot ○	Closed dot ●
Interval Notation	Parentheses ( )	Brackets [ ]	Parentheses ( )	Brackets [ ]

Solutions to Inequalities in inequality form:

Review:

< - means “less than” ≤ - means “greater than”

> - means “less than or equal to” ≥ - means “greater than or equal to”

To find the solution of an inequality, we solve for x. This means we will have to get x by itself.

Examples: x > 3 means the value of x is anything greater than 3.

x ≤ 5 means the value of x is anything less than or equal to 5.

Graphing solutions of inequalities:

To graph the solutions of inequalities, we will use a number line. For solutions where x is < or >, we will use an open dot ○ on the number. For solutions where x is ≤ or ≥, we will use a closed dot ● on the number. From there we draw an arrow from the dot to represent all values of x.

Examples: x > 3 x ≤ 5

SECTION 27 SOLVING LINEAR INEQUALITIES LINEAR INEQUALITY IN ONE

-3 -2 -1 0 1 2 3 4 5

-3 -2 -1 0 1 2 3 4 5 6

x ≥ 4 x < -1

SECTION 27 SOLVING LINEAR INEQUALITIES LINEAR INEQUALITY IN ONE

-3 -2 -1 0 1 2 3 4 5 6

Writing Solutions in Interval Notation:

To write a solution for x in interval notation, we use parentheses and/or brackets. If x is greater than or greater than or equal to a number, it goes towards infinity ∞. If x is less than or less than or equal to a number, it goes towards negative infinity -∞. We will always use parentheses on infinity and negative infinity.

Examples: x > 3 (3, ∞) x ≤ 5 (-∞, 5] x ≥ 4 [4, ∞) x < -1 (-∞, -1)

Addition Property of Inequality:

If a, b, and c are real numbers, then a < b and a + c < b + c are equivalent inequalities.

Example 1: Solve .

SECTION 27 SOLVING LINEAR INEQUALITIES LINEAR INEQUALITY IN ONE

Multiplication Property of Inequality:

1. If a, b, and c are real numbers, and c is positive, then a < b and ac < bc are equivalent inequalities.

2. If a, b, and c are real numbers, and c is negative, then a < b and ac > bc are equivalent inequalities.

What this means: When you multiply or divide by a negative while solving an inequality, you have to flip/change/reverse the inequality sign.

Example 2: Solve . Example 3: Solve .

SECTION 27 SOLVING LINEAR INEQUALITIES LINEAR INEQUALITY IN ONE

Example 4: Solve, graph, and write in interval notation: .

SECTION 27 SOLVING LINEAR INEQUALITIES LINEAR INEQUALITY IN ONE

-3 -2 -1 0 1 2 3 4 5

(-∞, 4]

Example 5: Solve, graph, and write in interval notation:

-3 -2 -1 0 1 2 3 4 5

SECTION 27 SOLVING LINEAR INEQUALITIES LINEAR INEQUALITY IN ONE

Compound Inequalities:

Inequalities containing one inequality symbol are called simple inequalities, while inequalities containing two inequality symbols are called compound inequalities. A compound inequality is two simple inequalities combined:

3 < x < 5 means 3 < x and x < 5

Read: “x is greater than three and less than five”.

Example 6: Solve, graph, and write in interval notation: .

SECTION 27 SOLVING LINEAR INEQUALITIES LINEAR INEQUALITY IN ONE

-3 -2 -1 0 1 2 3 4 5

Using inequalities with word problems:

Example 7: Mary has $350 to spend on clothes (exclude taxes). She plans on buying three pairs of jeans priced at $50 each and then several t-shirts priced at $15 dollars each. How many t-shirts can Mary buy along with the three pairs of jeans and stay within her budget?

x = the number of t-shirts

Solution: x ≤13.33, which means Mary can buy 13 t-shirts and three pairs of jeans for less than $350.

Example 8: Danny works at a local car dealership. He earns $2,000 a month plus 5% commission of the cars he sells. What is the value of the cars Danny must sell in a year to earn at least $65,000?

x = the value of the cars

Solution: x ≥ $820,000, which means Danny must sell $820,000 worth of cars during a year to earn at least $65,000 a year.

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