Question

How do you solve `|2x - 5| + 1< 16`?

Answer 1

See a solution process below:

Explanation:

First, subtract `color(red)(1)` from each side of the inequality to isolate the absolute value function while keeping the equation balanced:

`abs(2x - 5) + 1 - color(red)(1) < 16 - color(red)(1)`

`abs(2x - 5) + 0 < 15`

`abs(2x - 5) < 15`

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.

`-15 < 2x - 5 < 15`

Next, add `color(red)(5)` to each segment of the system of inequalities to isolate the `x` term while keeping the system balanced:

`-15 + color(red)(5) < 2x - 5 + color(red)(5) < 15 + color(red)(5)`

`-10 < 2x - 0 < 20`

`-10 < 2x < 20`

Now, divide each segment by `color(red)(2)` to solve for `x` while keeping the system balanced:

`-10/color(red)(2) < (2x)/color(red)(2) < 20/color(red)(2)`

`-5 < (color(red)(cancel(color(black)(2)))x)/cancel(color(red)(2)) < 10`

`-5 < x < 10`

Or

`x > -5` and `x < 10`

Or, in interval notation:

`(-5, 10)`

Answer 2

Given: `|2x - 5| + 1< 16`

Subtract 1 from both sides:

`|2x - 5| < 15`

The absolute value function has the piecewise definition:

`|f(x)| = {(f(x); f(x) >= 0),(-f(x); f(x) < 0):}`

This allows us to separate the inequality into two:

`2x - 5 < 15` and `-(2x - 5) < 15`

Multiply both sides of the second inequality by -1:

`2x - 5 < 15` and `2x - 5 > -15`

Add 5 to both sides of both inequalities:

`2x < 20` and `2x > -10`

Divide both sides of both inequalities by 2:

`x < 10` and `x > -5`

This can be written as:

`-5 < x < 10`