Question

How do you graph and solve `| 3 - 7 x |>17`?

Answer 1

See a solution process below: `(-2; 20/7)`

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.

We can rewrite and solve this inequality as:

`-17 > 3 - 7x > 17`

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

`-17 - color(red)(3) > 3 - color(red)(3) - 7x > 17 - color(red)(3)`

`-20 > 0 - 7x > 14`

`-20 > -7x > 14`

Now, divide each segment by `color(blue)(-7)` to solve for `x` while keeping the system balanced. Because we are multiplying or dividing an inequality by a negative number we must reverse the inequality operators.

`(-20)/color(blue)(-7) color(red)(<) (-7x)/color(blue)(-7) color(red)(<) 14/color(blue)(-7)`

`20/7 color(red)(<) (color(blue)(cancel(color(black)(-7)))x)/cancel(color(blue)(-7)) color(red)(<) -2`

`20/7 color(red)(<) x color(red)(<) -2`

Or.

`x < -2`; `x > 20/7`

Or, in interval notation:

`(-oo, -2)`; `(20/7, +oo)`

To graph this we will draw a vertical lines at `-2` and `20/7` on the horizontal axis.

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

We will shade outside the two lines to show the interval where the inequality is true.