Solved on Jan 20, 2024

Solve the equation $9-|5x+1|=9$ and provide the solution(s) as a comma-separated list. If there is no solution, enter "NO SOLUTION".

STEP 1

Assumptions
1. We are solving the equation $9 - |5x + 1| = 9$ .
2. The absolute value function $|a|$ returns the non-negative value of $a$ .
3. We will consider two cases for the absolute value: when the expression inside is non-negative and when it is negative.

STEP 2

First, we will simplify the equation by isolating the absolute value expression on one side.
$9 - |5x + 1| = 9$

STEP 3

Subtract 9 from both sides of the equation to isolate the absolute value.
$9 - |5x + 1| - 9 = 9 - 9$

STEP 4

Simplify both sides of the equation.
$-|5x + 1| = 0$

STEP 5

Multiply both sides of the equation by -1 to get the absolute value by itself.
$|5x + 1| = 0$

STEP 6

Now we consider the definition of absolute value. The absolute value of an expression is 0 if and only if the expression inside is equal to 0.

STEP 7

Set the expression inside the absolute value equal to 0.
$5x + 1 = 0$

STEP 8

Subtract 1 from both sides of the equation to solve for $x$ .
$5x + 1 - 1 = 0 - 1$

STEP 9

Simplify both sides of the equation.
$5x = -1$

STEP 10

Divide both sides of the equation by 5 to solve for $x$ .
$\frac{5x}{5} = \frac{-1}{5}$

STEP 11

Simplify both sides of the equation.
$x = -\frac{1}{5}$
Since the absolute value equation $|5x + 1| = 0$ has only one solution, where the expression inside the absolute value is zero, we conclude that the solution to the equation is $x = -\frac{1}{5}$ .
The solution to the equation $9 - |5x + 1| = 9$ is $x = -\frac{1}{5}$ .

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Solve the equation $9-|5x+1|=9$ and provide the solution(s) as a comma-separated list. If there is no solution, enter "NO SOLUTION".

STEP 1

Assumptions
1. We are solving the equation $9 - |5x + 1| = 9$ .
2. The absolute value function $|a|$ returns the non-negative value of $a$ .
3. We will consider two cases for the absolute value: when the expression inside is non-negative and when it is negative.

STEP 2

First, we will simplify the equation by isolating the absolute value expression on one side.
$9 - |5x + 1| = 9$

STEP 3

Subtract 9 from both sides of the equation to isolate the absolute value.
$9 - |5x + 1| - 9 = 9 - 9$

STEP 4

Simplify both sides of the equation.
$-|5x + 1| = 0$

STEP 5

Multiply both sides of the equation by -1 to get the absolute value by itself.
$|5x + 1| = 0$

STEP 6

Now we consider the definition of absolute value. The absolute value of an expression is 0 if and only if the expression inside is equal to 0.

STEP 7

Set the expression inside the absolute value equal to 0.
$5x + 1 = 0$

STEP 8

Subtract 1 from both sides of the equation to solve for $x$ .
$5x + 1 - 1 = 0 - 1$

STEP 9

Simplify both sides of the equation.
$5x = -1$

STEP 10

Divide both sides of the equation by 5 to solve for $x$ .
$\frac{5x}{5} = \frac{-1}{5}$

STEP 11

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Solve the equation 9−∣5x+1∣=99-|5x+1|=99−∣5x+1∣=9 and provide the solution(s) as a comma-separated list. If there is no solution, enter "NO SOLUTION".

STEP 1

Assumptions1. We are solving the equation 9−∣5x+1∣=99 - |5x + 1| = 99−∣5x+1∣=9.2. The absolute value function ∣a∣|a|∣a∣ returns the non-negative value of aaa.3. We will consider two cases for the absolute value: when the expression inside is non-negative and when it is negative.

STEP 2

First, we will simplify the equation by isolating the absolute value expression on one side.9−∣5x+1∣=99 - |5x + 1| = 99−∣5x+1∣=9

STEP 3

Subtract 9 from both sides of the equation to isolate the absolute value.9−∣5x+1∣−9=9−99 - |5x + 1| - 9 = 9 - 99−∣5x+1∣−9=9−9

STEP 4

Simplify both sides of the equation.−∣5x+1∣=0-|5x + 1| = 0−∣5x+1∣=0

STEP 5

Multiply both sides of the equation by -1 to get the absolute value by itself.∣5x+1∣=0|5x + 1| = 0∣5x+1∣=0

STEP 6

Now we consider the definition of absolute value. The absolute value of an expression is 0 if and only if the expression inside is equal to 0.

STEP 7

Set the expression inside the absolute value equal to 0.5x+1=05x + 1 = 05x+1=0

STEP 8

Subtract 1 from both sides of the equation to solve for xxx.5x+1−1=0−15x + 1 - 1 = 0 - 15x+1−1=0−1

STEP 9

Simplify both sides of the equation.5x=−15x = -15x=−1

STEP 10

Divide both sides of the equation by 5 to solve for xxx.5x5=−15\frac{5x}{5} = \frac{-1}{5}55x​=5−1​

STEP 11

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Solve the equation $9-|5x+1|=9$ and provide the solution(s) as a comma-separated list. If there is no solution, enter "NO SOLUTION".

Assumptions
1. We are solving the equation $9 - |5x + 1| = 9$ .
2. The absolute value function $|a|$ returns the non-negative value of $a$ .
3. We will consider two cases for the absolute value: when the expression inside is non-negative and when it is negative.

First, we will simplify the equation by isolating the absolute value expression on one side.
$9 - |5x + 1| = 9$

Subtract 9 from both sides of the equation to isolate the absolute value.
$9 - |5x + 1| - 9 = 9 - 9$

Simplify both sides of the equation.
$-|5x + 1| = 0$

Multiply both sides of the equation by -1 to get the absolute value by itself.
$|5x + 1| = 0$

Set the expression inside the absolute value equal to 0.
$5x + 1 = 0$

Subtract 1 from both sides of the equation to solve for $x$ .
$5x + 1 - 1 = 0 - 1$

Simplify both sides of the equation.
$5x = -1$

Divide both sides of the equation by 5 to solve for $x$ .
$\frac{5x}{5} = \frac{-1}{5}$