Solve the inequality |x+2|&lt;|12x−1|.

Question

Solve the inequality |x+2|&amp;lt;|12x−1|.

Wriedge · Accepted Answer

Step 1
The expression |x−a| is the absolute value of x-a. For

x \geq a

this is equal to x−a.
For x<a this is equal to a−x. Using this the given inequality can be solved by splitting it into several cases.
Step 2
The given inequality is

| x + 2 | < | \frac{1}{2} x - 1 |

. This can be written as

| x + 2 | < \frac{| x - 2 |}{2}

.
We will solve this inequality by taking three cases.
Case 1:

x \geq 2

For this case |x+2|=x+2 and

\frac{| x - 2 |}{2} = \frac{x - 2}{2}

.
Use this to solve the inequality for this case.

x + 2 < \frac{x - 2}{2}

2x+4<x-2
x<-6
x<-6 and

x \geq 2

are not both possible. So there are no solutions in this case.
Step 3
Case 2:

- 2 \leq x < 2

For this case |x+2|=x+2 and

\frac{| x - 2 |}{2} = \frac{2 - x}{2}

.
Use this to solve the inequality for this case.

x + 2 < \frac{2 - x}{2}

2x+4<2-x
3x<-2

x < - \frac{2}{3}

Combining

- 2 \leq x < 2 and x < - \frac{2}{3}

gives

x \in [- 2, - \frac{2}{3}]

.
Step 4
Case 3: x<−2
For this case |x+2|=-x-2 and

\frac{| x - 2 |}{2} = \frac{2 - x}{2}

.
Use this to solve the inequality for this case.

- x - 2 < \frac{2 - x}{2}

-2x-4<2-x
-4<2+x
-6<x
Combining this with x<−2 gives

x \in (- 6, - 2)

.
Hence, the solution to the given inequality is

x \in (- 6, - 2) \cup [- 2, - \frac{2}{3}) or x \in (- 6, - \frac{2}{3})

.

Solve the inequality |x+2|<|\frac{1}{2}x-1|.

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