Question

Solve absolute value inequalities |x+5|>-1 and b) |3-x|<5

Answer 1

|x+5| > -1 is true for all x-values.

|3-x| < 5 is true when -2 < x < 8.

Explanation:

Given the inequality:

`\displaystyle`

The inequality is true for all values of x. This is because no matter what real numbers/values you substitute in x-term, the output will always be positive, which is greater than negative.

Given another inequality for part b:

`\displaystyle`

This inequality can have a specific interval as the right side is greater than the negative. Therefore, apply the definition of absolute value:

For x ≥ 0,

`\displaystyle{3-x < 5}\\\\\displaystyle{3-5 < x}\\\\\displaystyle{-2 < x}\\\\\displaystyle{x > -2}`

For x < 0,

`\displaystyle{-(3-x) < 5}\\\\\displaystyle{-3+x < 5}\\\\\displaystyle{x < 8}`

Therefore, when combining both intervals, we have the following:

`\displaystyle{-2 < x < 8}`

Hence, the inequality is true for -2 < x < 8.