How do you solve the equation abs(4/3-2/3x)=3/4?

May 4, 2017

$x = \frac{25}{8}$ and $\frac{7}{8}$

Explanation:

Absolute value equations are a little tough. Some teachers say that they "make numbers positive", but that doesn't mean $x$ is always a positive number. Absolute value bars concern distance. That's why they "make the number positive"; because there's no such thing as a negative distance (we can't have $- 262$ feet).

When we have $\sqrt{x} = y$, we take the inverse of the square root to "undo" it. But what's the inverse of absolute value bars? Nothing. But, we do know that whatever $x$ equals, it can be either positive or negative.

So, the way we solve for absolute value equations is to let one equation be positive and another be negative:

$\left\mid \frac{4}{3} - \frac{2}{3} x \right\mid = \frac{3}{4}$

Positive situation
$\left(\frac{4}{3} - \frac{2}{3} x\right) = \frac{3}{4}$

subtract $\frac{4}{3}$ on both sides

$- \frac{2}{3} x = - \frac{7}{12}$

divide by $- \frac{2}{3}$

$x = \frac{7}{8}$

Negative solution
$- \left(\frac{4}{3} - \frac{2}{3} x\right) = \frac{3}{4}$

divide by $- 1$ on both sides

$\frac{4}{3} - \frac{2}{3} x = - \frac{3}{4}$

subtract $\frac{4}{3}$ on both sides

$- \frac{2}{3} x = - \frac{25}{12}$

divide by $- \frac{2}{3}$ on both sides

$x = \frac{25}{8}$

So, our solutions are $x = \frac{25}{8}$ and $\frac{7}{8}$. Just to double check, let's grahp our equation:

graph{y=abs(4/3-2/3x)-3/4}

Yep, intercepts at $0.875$ and $3.125$, or $\frac{7}{8}$ and $\frac{25}{8}$.

May 4, 2017

$x = \frac{7}{8} \text{ or } x = \frac{25}{8}$

Explanation:

$\text{the value of the "color(blue)"absolute value function}$ is always positive.

$\text{However, the value of the expression inside the bars}$
$\text{can be positive or negative}$

$\text{This means there are 2 possible solutions to the equation}$

$\textcolor{red}{\pm} \left(\frac{4}{3} - \frac{2}{3} x\right) = \frac{3}{4}$

$\textcolor{b l u e}{\text{First possible solution}}$

$\frac{4}{3} - \frac{2 x}{3} = \frac{3}{4} \leftarrow \textcolor{red}{\text{ positive value}}$

$\Rightarrow \frac{2 x}{3} = \frac{4}{3} - \frac{3}{4} = \frac{7}{12}$

$\Rightarrow 24 x = 21 \leftarrow \textcolor{red}{\text{ cross-multiplying}}$

$\Rightarrow x = \frac{21}{24} = \frac{7}{8} \leftarrow \textcolor{m a \ge n t a}{\text{ first possible solution}}$

$\textcolor{b l u e}{\text{Second possible solution}}$

$\frac{2 x}{3} - \frac{4}{3} = \frac{3}{4} \leftarrow \textcolor{red}{\text{ negative value}}$

$\Rightarrow \frac{2 x}{3} = \frac{3}{4} + \frac{4}{3} = \frac{25}{12}$

$\Rightarrow 24 x = 75 \leftarrow \textcolor{red}{\text{ cross-multiplying}}$

$\Rightarrow x = \frac{75}{24} = \frac{25}{8} \leftarrow \textcolor{m a \ge n t a}{\text{ second possible solution}}$

$\textcolor{b l u e}{\text{As a check}}$

Substitute these values into the left side of the equation and if equal to the right side then they are the solutions.

$| \frac{4}{3} - \left(\frac{2}{3} \times \frac{7}{8}\right) | = | \frac{4}{3} - \frac{7}{12} | = | \frac{3}{4} | = \frac{3}{4}$

$| \frac{4}{3} - \left(\frac{2}{3} \times \frac{25}{8}\right) | = | \frac{4}{3} - \frac{25}{12} | = | - \frac{3}{4} | = \frac{3}{4}$

$\Rightarrow x = \frac{7}{8} \text{ or " x=25/8" are the solutions}$