Solve the equation for all values of $x$ . $\newline$ $|4 x+2|-8=x$ $\newline$ Answer: $x=$

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Grade 8

Evaluate absolute value expressions

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Q. Solve the equation for all values of

x

\newline

|4 x+2|-8=x

\newline

Answer:

x=

Isolate absolute value: We have the equation $|4x+2|-8=x$ . To solve for $x$ , we first need to isolate the absolute value expression on one side of the equation. $\newline$ Add $8$ to both sides of the equation to isolate the absolute value. $\newline$ $|4x+2|-8+8=x+8$ $\newline$ $|4x+2|=x+8$
Split into two equations: Now we have $|4x+2|=x+8$ . The absolute value equation can be split into two separate equations because the expression inside the absolute value can be either positive or negative. $\newline$ The two cases are: $\newline$ $1$ . $4x+2 = x+8$ (when the expression inside the absolute value is positive or zero) $\newline$ $2$ . $4x+2 = -(x+8)$ (when the expression inside the absolute value is negative)
Solve first case: Solve the first case $4x+2 = x+8$ . $\newline$ Subtract $x$ from both sides: $\newline$ $4x - x + 2 = x - x + 8$ $\newline$ $3x + 2 = 8$ $\newline$ Subtract $2$ from both sides: $\newline$ $3x + 2 - 2 = 8 - 2$ $\newline$ $3x = 6$ $\newline$ Divide both sides by $3$ : $\newline$ $\frac{3x}{3} = \frac{6}{3}$ $\newline$ $x = 2$
Solve second case: Solve the second case $4x+2 = -(x+8)$ .
Distribute the negative sign on the right side:
$4x + 2 = -x - 8$
Add $x$ to both sides:
$4x + x + 2 = -x + x - 8$
$5x + 2 = -8$
Subtract $2$ from both sides:
$5x + 2 - 2 = -8 - 2$
$5x = -10$
Divide both sides by $5$ :
$\frac{5x}{5} = \frac{-10}{5}$
$4x + 2 = -x - 8$ $0$
Check solutions: We have found two potential solutions for the equation $|4x+2|-8=x$ : $x = 2$ and $x = -2$ . However, we must check these solutions in the original equation to ensure they do not create any contradictions, as sometimes absolute value equations can yield extraneous solutions. $\newline$ Check $x = 2$ : $\newline$ $|4(2)+2|-8=2$ $\newline$ $|8+2|-8=2$ $\newline$ $|10|-8=2$ $\newline$ $10-8=2$ $\newline$ $2=2$ (This is true, so $x = 2$ is a valid solution.) $\newline$ Check $x = -2$ : $\newline$ $x = 2$ $1$ $\newline$ $x = 2$ $2$ $\newline$ $x = 2$ $3$ $\newline$ $x = 2$ $4$ $\newline$ $x = 2$ $5$ (This is true, so $x = -2$ is also a valid solution.)