Exercise
Solve the inequality:
|x+2|+|x-2|\leq 10
Final Answer
Solution
|x+2|+|x-2|\leq 10
We check when the phrases in the absolute values equal zero:
x+2=0 \rightarrow x=-2
x-2=0 \rightarrow x=2
We divide the x-axis into foreign sections by the points we found. We get three sections:
x\leq -2, -2< x\leq 2, x>2
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 -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
-(x+2)-(x-2)\leq 10
We solve the inequality:
-x-2-x+2\leq 10
-2x\leq 10
x\geq -5
Now, intersect this result with the original section. Meaning,
x\geq -5
and
x\leq -2
Together we get
-5 \leq x\leq -2
Moving on to the second section:
-2< x\leq 2
We choose the number x = 0. 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
(x+2)-(x-2)\leq 10
Solve the inequality:
x+2-x+2\leq 10
4\leq 10
Hence, the inequality solution is all x.
Now, intersect this result with the original section:
-2< x\leq 2
together, we get
-2< x\leq 2
Lastly, the third section:
x>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
(x+2)+(x-2)\leq 10
Solve the inequality:
x+2+x-2\leq 10
2x\leq 10
x\leq 5
Now, intersect this result with the original section, meaning
x\leq 5
and
x>2
together we get
2< x\leq 5
The final step is to take all the solutions we received and union them, meaning
-5 \leq x\leq -2
or
-2< x\leq 2
or
2< x\leq 5
Hence, our final answer is
-5\leq x\leq 5
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