Question

|5x+8|=|10x+7| solve the equation.

Answer 1

We have to take care of absolute values. The absolute value of an expression is the positive version of that expression, i.e.

$|x| = \begin{cases} x &\text{ if } x \geq 0\\ -x &\text{ if } x < 0 \end{cases}$

So, first of all, we must observe that $5x+8$ is positive if

$5x+8 \geq 0 \iff 5x \geq -8 \iff x \geq \dfrac{-8}{5}$

and similarly,

$10x+7 \geq 0 \iff 10x \geq -7 \iff x \geq \dfrac{-7}{10}$

So, since

$\dfrac{-8}{5} < \dfrac{-7}{10}$, we can divide the number line in three zones:

Zone 1: x < -8/5

In this zone, both expressions are negative. This means that

$|5x+8|= -5x-8,\qquad |10x+7| = -10x-7$

In fact, as we already said, the absolute value flips the sign of an expression if that expression is negative. So, the equation becomes

$-5x-8 = -10x-7 \iff 5x=1 \iff x = \dfrac{1}{5}$

But we can't accept this solution (yet), because we're supposing x < -8/5.

Zone 2: -8/5 < x < -7/10

In this zone, 5x+8 has become positive, while 10x+7 is still negative. This means that

$|5x+8|= 5x+8,\qquad |10x+7| = -10x-7$

So, the equation becomes

$5x+8 = -10x-7 \iff 15x=-15 \iff x = -1$

Since indeed -8/5 < -1 < -7/10, we can accept this solution.

Zone 3: x > -7/10

In this zone, both expressions are positive, which means that the absolute value changes nothing:

$|5x+8|= 5x+8,\qquad |10x+7| = 10x+7$

So, the equation becomes

$5x+8 = 10x+7 \iff 5x=1 \iff x = \dfrac{1}{5}$

Since indeed 1/5 > -7/10, we can accept this solution.