Verified answer

|x - 5| ≤ 3

Definition of absolute value inequality.

-3 ≤ x - 5 ≤ 3

Add 5 to all three parts.

2 ≤ x ≤ 8

In interval notation, this solution is [2,8].

On a number line you would put closed dots on the points 2 and 8, then connect them with a line in between. Sort of like this:

<------------------------------>

.....2.....................8

The closed dots would go on the line where 2 and 8 are. Then you would draw a line connecting those dots.

The modulus sign means that no matter whether x-5 is positive or negative, you always take the positive value.

So you split this up into two equations where you know in one of then it'll be positive and the other negative.

below, "=<" means less than or equal to

x - 5 =< 3

5 - x =< 3

Notice that if x - 5 is positive that 5 - x, which is -(x-5), MUST be negative.

So,

x =< 3 + 5

x =< 8 is one of your lines. Its a "or equal to" so you draw a full vertical line going through "8" on the x axis.

and

-x =< -2

therefore

x => 2 .......... (Notice how the less than sign changed to a greater than sign because both sides were multiplied by a negative number)

so for this you draw a vertical line going through 2 on the x axis.

Now you have two lines, and x is greater than or equal to 2 and less than or equal to 8. So x can be 2, 8, or anything inbetween. So if you're asked to shade a valid domain or something, you can shade the area between those two lines.

- 3 ≤ x - 5 ≤ 3

2 ≥ x ≤ 8

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