Question

Solve 2lx-8l + 14 = 4x

Answer 1

x = 5

Explanation:

When solving an absolute-value equation, we can break it into two conditions:

`\displaystyle{|x-a| = \begin{cases} x-a \ \ \left(x\geq a\right) \\ -\left(x-a\right)} \ \ \left(x < a\right) \end{cases}}`

Therefore, when x ≥ 8, we have:

`\displaystyle{2\left(x-8\right)+14=4x}`

Distribute 2 in x - 8:

`\displaystyle{2x-16+14=4x}\\\\\displaystyle{2x-2=4x}\\\\\displaystyle{-2=4x-2x}\\\\\displaystyle{-2=2x}\\\\\displaystyle{x=-1}`

However, this solution is invalid because they will not be equal if you substitute x = -1 on both sides. The right side is negative while the left side is positive for all x-values.

Consider another interval where x < 8, we have:

`\displaystyle{-2\left(x-8\right)+14=4x}\\\\\displaystyle{-2x+16+14=4x}\\\\\displaystyle{-2x+30=4x}\\\\\displaystyle{30=4x+2x}\\\\\displaystyle{30=6x}\\\\\displaystyle{x=5}`

This solution is valid because both sides are equal when you substitute x = 5 in. Hence, the solution to the equation is x = 5.