Transcript 2.2 (M2) Solve Absolute Value Equations & Inequalities

Warm-Up
 |x |=3
 |x |= -7
 |2x |=10
 |x+3 |=6
2.2 (M2) Solve
Absolute Value
Equations &
Inequalities
Vocabulary
An extraneous solution is an
apparent solution that must
be rejected because it does
not satisfy the original
equation.
Example 1 – Solve an
absolute value equation
Solve | 2x-5 | = 9
Because this is an absolute value
equation, the expression within the
abs. value can equal 9 or -9. Set the
equation equal to 9 & to -9 and solve.
2x – 5 = 9
2x = 14
x=7
2x – 5 = -9
2x = -4
x = -2
The solutions are -2, 7.
Try these . . .
1. | x + 3 | = 7
2. | x – 2 | = 6
3. | 2x + 1 | = 9
Example 2 – Check for
extraneous solutions
Solve | x + 2 | = 3x.
Just like example 1, we will set the
expression within the abs. value to 3x
& to -3x. Make sure to check for
extraneous solutions.
x + 2 = 3x
x + 2 = -3x
2 = 2x
2 = -4x
x=1
x = -½
Substitute into original equation.
| 1 + 2 | = 3(1) | -½ + 2 | = 3(-½)
|3|=3
| 1½ | = -1½
3=3
1½ ≠ -1½
Therefore, x = 1 is the only
solution.
Example 3 – Solve an inequality
of the form |ax + b | > c
 This absolute value inequality is
equivalent to ax + b > c OR ax + b < -c.
 same
sign, same inequality
 opposite sign, opposite inequality
 Write the two inequalities, solve and
graph the solutions on a number line.
Solve | 2x – 1 | > 5.
Write as 2 inequalities.
2x – 1 > 5
2x > 6
x>3
OR
2x – 1 < -5
2x < -4
x < -2
Show solution on a number line.
Example 4 – Solve an inequality
of the form | ax + b | ≤ c
This absolute value
inequality can be rewritten as
a compound inequality.
Solve the compound
inequality and graph solution
on a number line.
Solve | 2x – 3 | ≤ 5.
Write as compound inequality.
-5 ≤ 2x – 3 ≤ 5
Add 3 to each expression.
-2 ≤ 2x ≤ 8
Divide each expression by 2.
-1 ≤ x ≤ 4
Graph solution on number line.
Try these . . .
1. | x + 7 | > 2
2. | 2x + 1 | ≥ 5
3. | x – 6 | ≤ 4
4. | x + 7 | < 2