Question

How do you solve `abs(9+x)<=7`?

Answer 1

See a solution process below:

Explanation:

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

`-7 <= 9 + x <= 7`

Subtract `color(red)(9)` from each segment of the system of inequalities to solve for `x` while keeping the system balanced:

`-color(red)(9) - 7 <= -color(red)(9) + 9 + x <= -color(red)(9) + 7`

`-16 <= 0 + x <= -2`

`-16 <= x <= -2`

Or

`x >= -16`; `x <= -2`

Or, in interval notation:

`[-16, -2]`

Answer 2

`-16<=x<=-2`

Explanation:

If `|a|<=b`, then `a<=b` and `a>=-b`.

So we can rewrite this problem as 2 equations:

`9+x<=7` and `9+x>=-7`

Now solve each equation and then use those to create your solution set.

`x<=7-9` and `x >= -7-9`

`x<=-2` and `x >= -16`

And finally, combine these two into a single equation which is your solution.

`-16<=x<=-2`

Final Answer