# Inequalities – Inequality with absolute value – Exercise 1862

Exercise

Solve the inequality:

$$|2x-1|>|x-1|$$

$$x<0 \text{ or } x>\frac{2}{3}$$

Solution

$$|2x-1|>|x-1|$$

We check when the phrases in the absolute values equal zero:

$$2x-1=0 \rightarrow x=\frac{1}{2}$$

$$x-1=0 \rightarrow x=1$$

We divide the x-axis into foreign sections by the points we found. We get three sections:

$$x\leq \frac{1}{2}, \frac{1}{2}< x\leq 1, x>1$$

In each section, we take the following steps:

1. Choose any number in the section.
2. Get rid of the absolute values ​​by the sign according to the number we chose, and solve the inequality.
3. Intersect result with the original section.

$$x\leq \frac{1}{2}$$

We choose the number x = -4. We set the number in our inequality and remove the absolute values by definition – If the result is positive, we simply remove the absolute value, and if the result is negative, we remove the absolute value and multiply by minus one. Hence, we get

$$-(2x-1)>-(x-1)$$

We solve the inequality:

$$-2x+1>-x+1$$

$$1-1>2x-x$$

$$0>x$$

Now, intersect this result with the original section. Meaning,

$$x<0$$

And

$$x\leq \frac{1}{2}$$

Together we get

$$x<0$$

Moving on to the second section:

$$\frac{1}{2}< x\leq 1$$

We choose the number

$$x=\frac{3}{4}$$

We set the number in our inequality and remove the absolute values by definition – If the result is positive, we simply remove the absolute value, and if the result is negative, we remove the absolute value and multiply by minus one. Hence, we get

$$2x-1>-(x-1)$$

Solve the inequality:

$$2x-1>-x+1$$

$$3x>2$$

$$x>\frac{2}{3}$$

Now, intersect this result with the original section, meaning

$$x>\frac{2}{3}$$

and

$$\frac{1}{2}< x\leq 1$$

together, we get

$$\frac{2}{3}< x\leq 1$$

Lastly, the third section:

$$x>1$$

We choose the number x = 4. We set the number in our inequality and remove the absolute values by definition – If the result is positive, we simply remove the absolute value, and if the result is negative, we remove the absolute value and multiply by minus one. Hence, we get

$$2x-1>x-1$$

Solve the inequality:

$$2x-x>1-1$$

$$x>0$$

Now, intersect this result with the original section, meaning

$$x>0$$

And

$$x>1$$

Together we get

$$x>1$$

The final step is to take all the solutions we received and union them, meaning

$$x<0$$

Or

$$\frac{2}{3}< x\leq 1$$

Or

$$x>1$$

$$x<0 \text{ or } x>\frac{2}{3}$$