Question

Find the real solutions of the equation. \[ |4 x+9|=5 \]

Answer 1

x = - 1 , x = - 3.5

Explanation:

the absolute value function always gives a positive value

however, the expression inside the bars can be positive or negative, that is

4x + 9 = 5 or - (4x + 9) = 5

solving both

4x + 9 = 5 ( subtract 9 from both sides )

4x = - 4 ( divide both sides by 4 )

x = - 1

or

-(4x + 9) = 5

- 4x - 9 = 5 ( add 9 to both sides )

- 4x = 14 ( divide both sides by - 4 )

x =
`(14)/(-4)` = - 3.5

As a check

substitute these values into the left side and if equal to the right side then they are solutions

x = - 1

| 4(- 1) + 9 | = | - 4 + 9 | = | 5 | = 5 ← true

x = - 3.5

| 4(- 3.5) + 9 | = | - 14 + 9 | = | - 5 | = 5 ← true

then solutions are x = - 1 , x = - 3.5