Absolute Value Equations and Inequalities

Absolute Value Equations and Inequalities
Absolute Value Equations of the form x
real number.
= a , where a is a non-negative
The solution is found by writing
x = a or x =− a.
Easy Example Solve x = 8
Solution We write x = 8orx=− 8 Answer
Harder Example Solve 2x + 4 = 8
Solution We write
2x + 4 = 8 or 2x+ 4= −8 and next we solve each equation for x.
−4 −4 −4 −4 and here we subtract 4 from both sides.
2x
= 4 or 2x
= −12 and next we divide both sides by 2.
So, x = 2 or x = −6 ← Answer
Absolute Value Inequalities of the form x
real number.
< a , where a is a non-negative
The solution is found by writing − a < x < a
Easy Example Solve x < 8
Solution We write − 8 < x < 8 Answer
Harder Example Solve 2x + 4 < 8
Solution We write
− 8 < 2x+ 4 < 8 and now we "isolate" the x.
−4 −4 −4 subtract 4 from each part of the compound inequality.
− 12 < 2x
< 4 and now divide by 2 in each part of the compound inequality.
− 6 < x < 2 ← Answer

Another Harder Example Solve 2x + 4 + 6 < 8
Solution We write
2x
+ 4 + 6 < 8 First we "isolate" the absolute value by subtracting 6 from
−6 −6 both sides of the inequality.
2x
+ 4 < 2
− 2 < 2x+ 4 < 2 and now we "isolate" the x.
−4 −4 − 4 subtract 4 from each part of the compound inequality.
− 6 < 2x
< −2 and now divide by 2 in each part of the compound inequality.
− 3 < x < −1 ← Answer
Absolute Value Inequalities of the form x
real number.
> a , where a is a non-negative
The solution is found by writing
x < − a or x > a
Easy Example Solve x > 8
Solution We write x 8 Answer
Harder Example Solve 2x + 4 > 8
Solution We write
2x+ 4 < − 8 or 2x+ 4 > 8 and now we "isolate" the x in each inequality.
−4 −4 −4 −4 subtract 4 from each side of each inequality.
2x < −12 or 2x > 4 and now divide by 2 on each side of each inequality.
x < −6 or x > 2 ← Answer
Another Harder Example Solve 6− x − 4 ≥ 5
Solution We write
6 −x
− 4 ≥ 5 First we "isolate" the absolute value by adding 4 to
+ 4 + 4 both sides of the inequality.
6−x
≥ 9
6 −x ≤ −9 or 6 −x ≥ 9 and now we "isolate" the x in each inequality.
−6 −6 −6 −6 subtract 6 from each side of each inequality.
−x ≤ −15 or −x ≥ 3 and now divide by −1 on each side of each inequality.
x ≥ 15 or x ≤ −3 ← Answer