Math

Question Solve the absolute value equation 11x+1052=13\frac{|11x+10|}{5}-2=13 for real value of xx.

Studdy Solution

STEP 1

Assumptions
1. We are given the equation 11x+1052=13\frac{|11x+10|}{5}-2=13.
2. We need to solve for the variable xx.

STEP 2

First, we want to isolate the absolute value expression. To do this, we can start by moving the constant term on the left side of the equation to the right side.
11x+105=13+2\frac{|11x+10|}{5} = 13 + 2

STEP 3

Now, add the numbers on the right side of the equation.
11x+105=15\frac{|11x+10|}{5} = 15

STEP 4

Next, we want to remove the fraction by multiplying both sides of the equation by 5.
11x+10=15×5|11x+10| = 15 \times 5

STEP 5

Multiply the numbers on the right side of the equation.
11x+10=75|11x+10| = 75

STEP 6

Now, we need to consider the two cases for the absolute value expression. The first case is when the expression inside the absolute value is positive, and the second case is when it is negative.
For the first case:
11x+10=7511x+10 = 75

STEP 7

Subtract 10 from both sides of the equation to isolate the term with the variable xx.
11x=751011x = 75 - 10

STEP 8

Now, subtract the numbers on the right side of the equation.
11x=6511x = 65

STEP 9

Divide both sides of the equation by 11 to solve for xx.
x=6511x = \frac{65}{11}

STEP 10

Calculate the value of xx.
x=6511=51011x = \frac{65}{11} = 5\frac{10}{11}
For the second case:
11x+10=7511x+10 = -75

STEP 11

Subtract 10 from both sides of the equation to isolate the term with the variable xx.
11x=751011x = -75 - 10

STEP 12

Now, subtract the numbers on the right side of the equation.
11x=8511x = -85

STEP 13

Divide both sides of the equation by 11 to solve for xx.
x=8511x = \frac{-85}{11}

STEP 14

Calculate the value of xx.
x=8511=7811x = \frac{-85}{11} = -7\frac{8}{11}

STEP 15

Now we have two possible solutions for xx from the two cases of the absolute value expression.
The solutions are x=51011x = 5\frac{10}{11} and x=7811x = -7\frac{8}{11}.

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