QuestionSolve the equation $2x - 3 = |x - 2|$ .

Studdy Solution

STEP 1

Assumptions1. The equation given is $x -3 = |x -|$ . The absolute value of a number is its distance from zero on the number line, which is always positive or zero.
3. The equation can be split into two cases, one for $x - \geq0$ and one for $x - <0$ .

STEP 2

First, we'll solve for the case where $x -2 \geq0$ . This means the absolute value can be removed.
$2x - = x -2$

STEP 3

To isolate $x$ , subtract $x$ from both sides of the equation.
$2x - x = -3 +2$

STEP 4

implify the equation to find the value of $x$ .
$x = -1$

STEP 5

Now, we need to check if the solution $x = -1$ satisfies the condition $x -2 \geq0$ . If it does not, then it is not a valid solution.
$-1 -2 \geq0$ This is not true, so $x = -1$ is not a solution.

STEP 6

Next, we'll solve for the case where $x -2 <0$ . This means the absolute value becomes the negative of the expression inside.
$2x -3 = -(x -2)$

STEP 7

implify the right side of the equation.
$2x -3 = -x +2$

STEP 8

To isolate $x$ , add $x$ to both sides of the equation.
$2x + x =2 +3$

STEP 9

implify the equation to find the value of $x$ .
$3x =5$ $x = \frac{5}{3}$

STEP 10

Now, we need to check if the solution $x = \frac{5}{3}$ satisfies the condition $x -2 <0$ . If it does not, then it is not a valid solution.
$\frac{5}{3} -2 <0$ This is true, so $x = \frac{5}{3}$ is a solution.
The solution to the equation $2x -3 = |x -2|$ is $x = \frac{5}{3}$ .

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