Question

How do you solve `-5\geq - 2x + 11\geq - 9`?

Answer 1

See a solution process below:

Explanation:

First, subtract `color(red)(11)` from each segment of the system of inequalities to isolate the `x` term while keeping the system balanced:

`-5 - color(red)(11) >= -2x + 11 - color(red)(11) >= -9 - color(red)(11)`

`-16 >= -2x + 0 >= -20`

`-16 >= -2x >= -20`

Now, divide each segment by `color(blue)(-2)` to solve for `x` while keeping the system balanced. However, because we are multiplying or dividing an inequality by a negative numbers we must also reverse the inequality operators:

`(-16)/color(blue)(-2) color(red)(<=) (-2x)/color(blue)(-2) color(red)(<=) (-20)/color(blue)(-2)`

`8 color(red)(<=) (color(blue)(cancel(color(black)(-2)))x)/cancel(color(blue)(-2)) color(red)(<=) 10`

`8 color(red)(<=) x color(red)(<=) 10`

Or

`x >= 8` and `x <= 10`

Or, in interval notation:

`[8, 10]`

Answer 2

Subtract 11 from everything
Divide everything by -2 (remember to "flip" the inequalities)

Explanation:

Given: `-5>=- 2x + 11>= - 9`

Subtract 11 from everything:

`-16>=- 2x>= -20`

Divide everything by -2:

`8<=x<= 10`