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Inequalities – Inequality with absolute value – Exercise 1852

Exercise

Solve the inequality:

|x+2|+|x-2|\leq 10

Final Answer


-5\leq x\leq 5

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:

  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 -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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