Exercise
Solve the inequality:
|2x-1|>|x-1|
Final Answer
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:
- Choose any number in the section.
- Get rid of the absolute values by the sign according to the number we chose, and solve the inequality.
- 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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