Question

How do you solve `3 abs( x + 5 ) < 21`?

Answer 1

`x<2` or `x> -12`

Solution

`3abs(x+5)<21`

`abs(x+5)<21/3`

`abs(x+5)<7`

It has two solutions

Either

`x+5<7` ......(i)

Or

`x+5> -7`.....(ii)

Solving (i)

`x+5-5<7-5`

`x+cancel5-cancel5<2`

`x<2`

Now, on solving (ii)

`x+5-5> -7-5`

`x+cancel5-cancel5> -12`

`x> -12`

Answer 2

We can divide both sides by 3 to get

`abs(x+5) < 21/3`

`->abs(x+5) < 7`

This tells us that `x+5` will lie between `-7 and 7`:

`-7 < x + 5 < 7`

`- 7 - 5 < x + cancel(5) - cancel(5) < 7 - 5`

`color(green)( -12 < x < 2`

To verify your answer , choose appropriate values of `x` and see if the inequality is satisfied

• Say `x = -10`

Left hand side = `3*abs(-10+5) = 3*abs(-5) = 3*5 = 15` (`<21`)

• Say `x = 1`

Left hand side = `3*abs(1+5) = 3*abs(6) = 3*6 = 18` (`<21`)