Question

How do you solve and write the following in interval notation: `5(x − 3) ≤ −2(x + 1)`?

Answer 1

See a solution process below:

Explanation:

To solve this, first, expand the terms in parenthesis by multiplying each term with the parenthesis by the term outside the parenthesis being careful to manage the signs correctly:

`color(red)(5)(x - 3) <= color(blue)(-2)(x + 1)`

`(color(red)(5) xx x) - (color(red)(5) xx 3) <= (color(blue)(-2) xx x) + (color(blue)(-2) xx 1)`

`5x - 15 <= -2x + (-2)`

`5x - 15 <= -2x - 2`

Next, add `color(red)(15)` and `color(blue)(2x)` to each side of the inequality to isolate the `x` term while keeping the inequality balanced:

`5x + color(blue)(2x) - 15 + color(red)(15) <= -2x + color(blue)(2x) - 2 + color(red)(15)`

`(5 + color(blue)(2))x - 0 <= 0 + 13`

`7x <= 13`

Now, divide each side of the inequality by `color(red)(7)` to solve for `x` while keeping the inequality balanced:

`(7x)/color(red)(7) <= 13/color(red)(7)`

`(color(red)(cancel(color(black)(7)))x)/cancel(color(red)(7)) <= 13/7`

`x <= 13/7`

We can write this in interval notation as:

`(-oo, 13/7]`