Solve the equation for all values of $x$ . $\newline$ $|5 x-4|-4=3 x$ $\newline$ Answer: $x=$

Full solution

Q. Solve the equation for all values of

x

\newline

|5 x-4|-4=3 x

\newline

Answer:

x=

Isolate Absolute Value: We are given the equation $|5x - 4| - 4 = 3x$ . To solve for $x$ , we first need to isolate the absolute value expression on one side of the equation. $\newline$ Add $4$ to both sides of the equation to isolate the absolute value. $\newline$ $|5x - 4| - 4 + 4 = 3x + 4$ $\newline$ $|5x - 4| = 3x + 4$
Consider Two Cases: Now we have to consider the two cases for the absolute value expression, because $|a| = b$ implies that $a = b$ or $a = -b$ .
Case $1$ : $5x - 4 = 3x + 4$
Case $2$ : $5x - 4 = -(3x + 4)$
Solve Case $1$ : Let's solve Case $1$ first. $\newline$ $5x - 4 = 3x + 4$ $\newline$ Subtract $3x$ from both sides to get the x terms on one side. $\newline$ $5x - 3x - 4 = 3x - 3x + 4$ $\newline$ $2x - 4 = 4$ $\newline$ Add $4$ to both sides to isolate the x term. $\newline$ $2x - 4 + 4 = 4 + 4$ $\newline$ $2x = 8$ $\newline$ Divide both sides by $2$ to solve for x. $\newline$ $\frac{2x}{2} = \frac{8}{2}$ $\newline$ x = $4$
Solve Case $2$ : Now let's solve Case $2$ . $\newline$ $5x - 4 = -(3x + 4)$ $\newline$ Distribute the negative sign on the right side. $\newline$ $5x - 4 = -3x - 4$ $\newline$ Add $3x$ to both sides to get the x terms on one side. $\newline$ $5x + 3x - 4 = -3x + 3x - 4$ $\newline$ $8x - 4 = -4$ $\newline$ Add $4$ to both sides to isolate the x term. $\newline$ $8x - 4 + 4 = -4 + 4$ $\newline$ $8x = 0$ $\newline$ Divide both sides by $8$ to solve for x. $\newline$ $\frac{8x}{8} = \frac{0}{8}$ $\newline$ x = $0$