Question

How do you graph and solve `|2x+3|<=15`?

Answer 1

See a solution process below:

Explanation:

The absolute value function takes any term and transforms it to its non-negative form. Therefore, we must solve the term within the absolute value function for both its negative and positive equivalent.

`-15 <= 2x + 3 <= 15`

First, subtract `color(red)(3)` from each segment of the system of equations to isolate the `x` term while keeping the system balanced:

`-15 - color(red)(3) <= 2x + 3 - color(red)(3) <= 15 - color(red)(3)`

`-18 <= 2x + 0 <= 12`

`-18 <= 2x <= 12`

Now, divide each segment of the system by `color(red)(2)` to solve for `x` while keeping the system balanced:

`-18/color(red)(2) <= (2x)/color(red)(2) <= 12/color(red)(2)`

`-9 <= (color(red)(cancel(color(black)(2)))x)/cancel(color(red)(2)) <= 6`

`-9 <= x <= 6`

Or

`x >= -9`; `x <= 6`

Or. in interval notation

`[-9, 6]`

To graph this we will draw vertical lines at `-9` and `6` on the horizontal axis.

The lines will be a solid line because the inequality operators contain "or equal to" clauses.

We will shade between the lines to show the interval: