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Question Solve the absolute-value equation 5∣12x+52∣−8=75\left|\frac{1}{2} x+\frac{5}{2}\right|-8=7 and graph the solutions.

Studdy Solution

STEP 1

Assumptions1. We are solving the absolute value equation 5∣1x+5∣−8=75\left|\frac{1}{} x+\frac{5}{}\right|-8=7. . The absolute value of a number is its distance from zero on the number line, and is always positive or zero.
3. The solutions to the equation are the x-values that make the equation true.

STEP 2

First, we need to isolate the absolute value expression. We can do this by adding8 to both sides of the equation.
5∣12x+52∣−8+8=7+85\left|\frac{1}{2} x+\frac{5}{2}\right|-8+8=7+8

STEP 3

implify the equation.
5∣12x+52∣=155\left|\frac{1}{2} x+\frac{5}{2}\right|=15

STEP 4

Next, divide both sides of the equation by to isolate the absolute value expression.
∣12x+2∣=15\frac{\left|\frac{1}{2} x+\frac{}{2}\right|}{}=\frac{15}{}

STEP 5

implify the equation.
∣12x+52∣=3\left|\frac{1}{2} x+\frac{5}{2}\right|=3

STEP 6

An absolute value equation can be split into two separate equations. One where the expression inside the absolute value is equal to the positive value on the other side of the equation, and one where it is equal to the negative of that value. This gives us two equations to solve.
12x+52=3\frac{1}{2} x+\frac{5}{2}=312x+52=−3\frac{1}{2} x+\frac{5}{2}=-3

STEP 7

olve the first equation by subtracting 52\frac{5}{2} from both sides.
12x+52−52=3−52\frac{1}{2} x+\frac{5}{2}-\frac{5}{2}=3-\frac{5}{2}

STEP 8

implify the equation.
12x=12\frac{1}{2} x=\frac{1}{2}

STEP 9

Multiply both sides of the equation by2 to solve for x.
2⋅2x=2⋅22 \cdot \frac{}{2} x=2 \cdot \frac{}{2}

STEP 10

implify the equation to find the first solution.
x=x=

STEP 11

Now, solve the second equation by subtracting 5\frac{5}{} from both sides.
x+5−5=−3−5\frac{}{} x+\frac{5}{}-\frac{5}{}=-3-\frac{5}{}

STEP 12

implify the equation.
2x=−112\frac{}{2} x=-\frac{11}{2}

STEP 13

Multiply both sides of the equation by2 to solve for x.
2⋅2x=2⋅−1122 \cdot \frac{}{2} x=2 \cdot -\frac{11}{2}

STEP 14

implify the equation to find the second solution.
x=−11x=-11The solutions to the equation are x=x= and x=−11x=-11.

STEP 15

To graph the solutions, plot the points x=x= and x=−11x=-11 on the number line.

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