Absolute value Equations and Inequalities

Question

Solve: 2|4x−5|+12≥20

Answer 1

|4x-5|≥(20-12)/2=4

4x-5≥4 or-4

4x≥9

≥x9/4

x≥2 1/4

x is also ≤1/4

in interval notation (∞,1/4]U[9/4,∞)

Answer 2

I am going to set these equal to 20 to find the critical values. (There are different ways you can solve this) First isolate the absolute values.

2|4x−5|+12 = 20 Subtract 12 from both sides

2|4x−5|+12 -12 = 20 -12 Simplify

2|4x−5| = 8 Divide by 2

2|4x−5| /2 = 8/2 Simplify

|4x−5| = 4 Now, split into 2 possible solutions

4x−5 = 4 4x−5 = -4 Add 5 to both sides of the equations

4x−5+5 = 4+5 4x−5 +5 = -4 + 5 Simplify

4x/4 = 9/4 4x/4 = 1/4 Divide both sides by 4

x = 9/4 x = 1/4 Since the original equation is ≥, this is an OR. So the final solution is:

(-∞,1/4] U [9/4,∞)