4.1 Solving Linear Inequalities

Objective: Solve and graph simple and compound inequalities in one variable.

What are inequalities?

Inequality Symbols

• Greater than: >

• Less than: <

• Greater than or equal to: ≥

• Less than or equal to: ≤

Inequalities

• Inequalities such as x ≤ 1 and p – 3 > 7 are linear inequalities in one variable.

• A solution of an inequality in one variable is a value of the variable that makes the inequality true.• Example: -4, 0.7, and 1 are solutions of x ≤ 1

• Two inequalities are equivalent if they have the same solutions

Properties of Inequalities

• To write an equivalent inequality:

• Add the same number to each side.• Subtract the same number from each side.• Multiply or divide each side by the same positive number.

• Multiply or divide each side by the same negative number and reverse the inequality symbol.

Solve the inequality.

• x – 4 > -6

• -5y + 2 ≥ -13

Solve the inequality.

• 7 – 4x < 1 – 2x

• 2x – 3 > x

Solve

• -x + 3 ≤ -6

• 3y – 5 < 10

• x + 3 < 8

• 2x – 3 > x

Graphing Inequalities

• The graph of an inequality in one variable consists of all points on a real number line that are solutions of the inequality.

• To graph an inequality in one variable:• Use an open dot () for < or >• Use a solid dot () for ≤ or ≥

Graphing Inequalities

• Graph x < 2

• Graph x ≥ 1

Solve the inequality and then graph your solution.

• 4x + 3 ≤ 6x – 5

• -x + 2 < -3

Compound Inequalities

AND

All real numbers greater than or equal to -2 and less than 1 can be written as:

-2 ≤ x < 1

Graph:

OR

All real numbers less than -1 or greater than or equal to 2 can be written as:

x < -1 or x ≥ 2

Graph:

A compound inequality is two simple inequalities joined by the word “and” or the word “or”

Graph the Compound Inequality

• x < -2 or x ≥ 3

• x ≤ -2 or x > 3

• -2 < x ≤ 3