Compound Inequalities Section 3.4 Compound Inequalities

Objectives Basic Concepts Symbolic Solutions and Number Lines Numerical and Graphical Solutions Interval Notation

Basic Concepts A compound inequality consists of two inequalities joined by the words and or or. 2x > –5 and 2x ≤ 8 x + 3 ≥ 4 or x – 2 < –6

Example Determine whether the given x-values are solutions to the compound inequalities. x + 2 < 7 and 2x – 3 > 3 x = 4, –4 Solution x + 2 < 7 and 2x – 3 > 3 Substitute 4 into the given compound inequality. 4 + 2 < 7 and 2(4) – 3 > 3 6 < 7 and 5 > 3 True and True Both inequalities are true, so 4 is a solution.

Example (cont) Determine whether the given x-values are solutions to the compound inequalities. x + 2 < 7 and 2x – 3 > 3 x = 4, –4 Solution x + 2 < 7 and 2x – 3 > 3 Substitute –4 into the given compound inequality. –4 + 2 < 7 and 2(–4) – 3 > 3 – 2 < 7 and –11 > 3 True and False To be a solution both inequalities must be true, so –4 is not a solution.

Symbolic Solutions and Number Lines We can use a number line to graph solutions to compound inequalities, such as x < 7 and x > –3. x < 7 x > –3 x < 7 and x > –3 Note: A bracket, either [ or ] or a closed circle is used when an inequality contains ≤ or ≥. A parenthesis, either ( or ), or an open circle is used when an inequality contains < or >.

Example Solve 3x + 6 > 12 and 5 – x < 11 . Graph the solution. Solution 3x + 6 > 12 and 5 – x < 11

Example Solve each inequality. Graph each solution set. Write the solution in set-builder notation. a. b. c. Solution a. b.

Example (cont) c.

Example Solve x + 3 < –2 or x + 3 > 2 Solution x + 3 < –2 or x + 3 > 2 x < –5 or x > –1

Example Write each expression in interval notation. a. –3 ≤ x < 7 b. x ≥ 4 c. x < –3 or x ≥ 5 d. {x|x > 0 and x ≤ 5} e. {x|x ≤ 2 or x ≥ 5}

Example Solve 2x + 3 ≤ –3 or 2x + 3 ≥ 5 Solution 2x + 3 ≤ –3 or 2x + 3 ≥ 5 2x ≤ –6 or 2x ≥ 2 x ≤ –3 or x ≥ 1 The solution set may be written as (, 3]  [1, )