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Inequalities – Inequality with absolute value – Exercise 1862

Exercise

Solve the inequality:

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

Final Answer


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.

We start with the first 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

Hence, our final answer is

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

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