Question

How do you solve and graph `2x < 10` and `-5x <5`?

Answer 1

See a solution process below:

Explanation:

To solve the first inequality, divide each side of the inequality by `color(red)(2)` to solve for `x` while keeping the inequality balanced:

`(2x)/color(red)(2) < 10/color(red)(2)`

`(color(red)(cancel(color(black)(2)))x)/cancel(color(red)(2)) < 5`

`x < 5`

To solve the second inequality, divide each side of the inequality by `color(blue)(-5)` to solve for `x` while keeping the inequality balanced. However, because we are multiplying or dividing an inequality by a negative term we must reverse the inequality operator:

`(-5x)/color(blue)(-5) color(red)(>) 5/color(blue)(-5)`

`(color(blue)(cancel(color(black)(-5)))x)/cancel(color(blue)(-5)) color(red)(>) -1`

`x > -1`

The solution is: `x > -1` and `x < 5`

Or, in interval notation: `(-1, 5)`

To graph this we will draw two vertical lines at `-1` and `5` on the horizontal axis.

The lines will be dashed lines because the inequality operators do not contain an "or equal to" clause.

We will shade between the interval of the lines: