QuestionSolve and graph the solution for $|2(x-2)|=9$ . Choose A for answers or B if the solution set is empty.

Studdy Solution

STEP 1

Assumptions1. The absolute value of a number is its distance from zero on the number line, regardless of direction. Therefore, it is always non-negative. . The equation to solve is $|(x-)|=9$ .

STEP 2

The absolute value equation $|2(x-2)|=9$ can be split into two separate equations. This is because the expression inside the absolute value brackets can be either positive or negative, and still yield the same absolute value.
$2(x-2)=9 \quad \text{and} \quad2(x-2)=-9$

STEP 3

Now, we solve the first equation $2(x-2)=9$ .
First, distribute the2 inside the parentheses.
$2x - =9$

STEP 4

Next, add4 to both sides of the equation to isolate the term with x.
$2x -4 +4 =9 +4$ $2x =13$

STEP 5

Finally, divide both sides of the equation by2 to solve for x.
$x = \frac{13}{2}$

STEP 6

Now, we solve the second equation $2(x-2)=-9$ .
First, distribute the2 inside the parentheses.
$2x -4 = -9$

STEP 7

Next, add4 to both sides of the equation to isolate the term with x.
$2x -4 +4 = -9 +4$ $2x = -5$

STEP 8

Finally, divide both sides of the equation by2 to solve for x.
$x = \frac{-5}{2}$

STEP 9

The solutions to the equation $|2(x-2)|=9$ are $x = \frac{13}{2}$ and $x = \frac{-5}{2}$ .

STEP 10

To graph the solution set, plot the points $\frac{13}{2}$ and $\frac{-5}{2}$ on a number line. These points are the only solutions to the equation, so they form the solution set.
The solution is $\left(\frac{-5}{2}, \frac{13}{2}\right)$ .

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