Math Snap

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.

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