Solving Compound and Absolute Value Inequalities

Chapter 1 – Section 6

Compound Inequalities Compound Inequality – a pair of inequalities joined by

and or or

Ex: -1 < x and x ≤ 3 which can be written as -1 < x ≤ 3

x < -1 or x ≥ 3 For and statements the value must satisfy both

inequalities For or statements the value must satisfy one of the

inequalities

And Inequalities

a) Graph the solution of 3x – 1 > -28 and 2x + 7 < 19.

3x > -27 and 2x < 12 x > -9 and x < 6

And Inequalities

b)Graph the solution of -8 < 3x + 1 <19-9 < 3x < 18-3 < x < 6

Or InequalitiesOr InequalitiesALGEBRA 2 LESSON 1-4ALGEBRA 2 LESSON 1-4

Graph the solution of 3x + 9 < –3 or –2x + 1 < 5.

3x + 9 < –3 or –2x + 1 < 5

3x < –12 –2x < 4

x < –4 or x > –2

Try These Problemsa) Graph the solution of 2x > x + 6 and x – 7 < 2

a) x > 6 and x < 9

b) Graph the solution of x – 1 < 3 or x + 3 > 8a) x < 4 or x > 11

Absolute Value Inequalities

Let k represent a positive real number │x │ ≥ k is equivalent to x ≤ -k or x ≥ k

│x │ ≤ k is equivalent to -k ≤ x ≤ k

Remember to isolate the absolute value before rewriting the problem with two inequalities

Solve |2x – 5| > 3. Graph the solution.

|2x – 5| > 3

2x – 5 < –3 or 2x – 5 > 3 Rewrite as a compound inequality.

x < 1 or x > 4

2x < 2 2x > 8 Solve for x.

Try This Problem

Solve │2x - 3 │ > 7

2x – 3 > 7 or 2x – 3 < -7

2x > 10 or 2x < -4

x > 5 or x < -2

Solve –2|x + 1| + 5 –3. Graph the solution.>–

|x + 1| 4 Divide each side by –2 and reverse the inequality.

<–

–2|x + 1| + 5 –3>–

–2|x + 1| –8Isolate the absolute value expression. Subtract 5 from each side.

>–

–4 x + 1 4Rewrite as a compound inequality.

<– <–

–5 x 3Solve for x.

<– <–

Try This Problem

Solve |5z + 3| - 7 < 34. Graph the solution. |5z + 3| -7 < 34 |5z + 3| < 41-41 < 5z + 3 < 41-44 < 5z < 38-44/5 < z < 38/5

-8 4/5 < z < 7 3/5

Homework

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