4.5 Solving Absolute ValueInequalities
11/16/12
Solve |x| < 3
What value of x would make what’s inside | | 7 or
-7 What numbers are less than 3 units away from 0?
Answer: Numbers to the right of -3, like -2, -1 and anything to the left of 3 like 2, 1
Mathematically written: -3 < x < 3
x > -3
x < 3
Solve |x| ≤ 3
Answer: Numbers to the right of -3 including 3 and anything to the left of 3 including 3.
Mathematically written: -3 ≤ x ≤ 3
x ≥ -3
x ≤ 3
Solve |x| > 4
What value of x would make what’s inside | | 7 or
-7 What numbers are more than 4 units away from 0?
Answer: Numbers to the right of 4, like 5, 6, 7 or anything to the left of - 4 like -5, -6, -7
Mathematically written: x < -4 or x > 4
x > 4
x < -4
Solve |x| ≥ 4
Answer: Anything to the right of 4 including 4 or anything to the left of - 4 including -4
Mathematically written: x ≤ -4 or x ≥ 4
x ≥ 4
x ≤ -4
Summary:Inequality Equivalent Form Graph
|x| < 3 -3 < x < 3 Shade between -3 and 3
|x| ≤ 3 -3 ≤ x ≤ 3
|x| > 4 x > 4 or x < -4 Shade opposite direction (away from each other) from 4 and -4|x| ≥ 4 x ≥ 4 or x ≤ -4
< and ≤ are both AND problems> and ≥ are both OR problems (think greater pronounced as greatOR)
“AND”
“OR”
Solving more complex Inequality
Example 1 Solve an Inequality of the Form
Solve . Then graph the solution.
+x b ≤ c
+x 4 ≤ 10
– +x 4 ≤ 10 Write equivalent compound inequality.10 ≤
– x ≤ 6 Subtract 4 from each expression.14 ≤
16–.14– 12– 10– 8– 6– 4– 2– 0 2 4 6 8
.
-4 -4 -4
SOLUTION
Checkpoint
Solve the inequality. Then graph your solution.
Solve an Absolute Value Inequality
1. +x 1 ≤ 4
ANSWER – x ≤ 35 ≤
6– 4– 2– 0 2 4. .3
7 < x < 9ANSWER
Example 2 Solve an Inequality of the Form
Solve . Then graph the solution.
+ax b c<
+2x 3 7<
– Write equivalent compound inequality.+2x 3 7<7 <
– Subtract 3 from each expression.10 2x 4<<
– Divide each expression by 2.5 x 2<<
6– 5– 4– 3– 0 1 22– 1– 3
SOLUTION
-3 -3 -3
2 2 2
Checkpoint
Solve the inequality. Then graph your solution.
Solve an Absolute Value Inequality
4. 2x 1 9<–
5. 1 4x 3<+ANSWER
1– 0..
– x 0.51 < <
1
ANSWER – x 54 < <
4– 2– 0 2 4
5
6
Example 3 Solve an Inequality of the Form
Solve . Then graph the solution.
+ax b c≥
x 1 32
1– ≥
FIRST INEQUALITY SECOND INEQUALITY
x 1 32
1– ≤ – x 1 3
2
1≥–
Add 1 to each side.x 2
2
1≤ – 4x
2
1≥
Checkpoint
Solve the inequality. Then graph your solution.
Solve an Absolute Value Inequality
7. 2x 3 7+ >
8. 4x 1 5+ ≥
ANSWER –1.5 orx x 1≥
1– 0 12–. .
≤
ANSWER < –5 orx 2>x
4– 2– 0 2 4 66–
5–
Homework: 4.5 p.201 #11-20 all
22-28 even34-44 even