QuestionSolve the equation $|x+8|=|2x+1|$ .

Studdy Solution

STEP 1

Assumptions1. The absolute value function is defined as $|a| = a$ for $a \geq0$ and $|a| = -a$ for $a <0$ . . The equation $|x+8|=|x+1|$ has to be solved for $x$ .

STEP 2

The equation $|x+8|=|2x+1|$ can be split into two cases based on the definition of absolute value functionCase1 $x+8 =2x+1$ when $x+8 \geq0$ and $2x+1 \geq0$
Case2 $x+8 = -(2x+1)$ when $x+8 <0$ and $2x+1 <0$

STEP 3

olve the equation for Case1 $x+8 =2x+1$
Subtract $x$ from both sides to get $8 = x +1$

STEP 4

Subtract1 from both sides to solve for $x$ $x =8 -1 =7$

STEP 5

Check if $x =7$ satisfies the conditions for Case1 $x+8 \geq0$ and $2x+1 \geq0$
Substitute $x =7$ into these inequalities $7+8 \geq0$ and $2(7)+1 \geq0$
Both inequalities are true, so $x =7$ is a valid solution for the equation.

STEP 6

olve the equation for Case2 $x+8 = -(2x+1)$ This simplifies to $x +8 = -2x -1$

STEP 7

Add $2x$ to both sides to get $3x + = -1$

STEP 8

Subtract8 from both sides to solve for $x$ $3x = -1 -8 = -$

STEP 9

Divide both sides by3 to solve for $x$ $x = -9 /3 = -3$

STEP 10

Check if $x = -3$ satisfies the conditions for Case2 $x+8 <0$ and $2x+ <0$
Substitute $x = -3$ into these inequalities $-3+8 <0$ and $2(-3)+ <0$
Both inequalities are false, so $x = -3$ is not a valid solution for the equation.

STEP 11

The solutions to the equation $|x+8|=|x+|$ are $x =7$ .

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