Solved on Dec 13, 2023

Solve the absolute value equation $|2x-1| = 9$ or indicate if it has no solution.

STEP 1

Assumptions
1. We are given the absolute value equation $|2x - 1| = 9$ .
2. We need to find all values of $x$ that satisfy the equation.

STEP 2

Understand that the absolute value of a number is its distance from zero on the number line, which means it is always non-negative. The absolute value equation $|2x - 1| = 9$ tells us that the expression $2x - 1$ is 9 units away from zero on the number line.

STEP 3

Recognize that since $|2x - 1|$ can be either positive or negative inside the absolute value, we need to consider two separate cases to solve for $x$ .

STEP 4

Set up the first case where the expression inside the absolute value is positive:
$2x - 1 = 9$

STEP 5

Solve the equation from STEP_4 for $x$ by first adding 1 to both sides of the equation:
$2x = 9 + 1$

STEP 6

Now, simplify the right side of the equation:
$2x = 10$

STEP 7

Divide both sides of the equation by 2 to solve for $x$ :
$x = \frac{10}{2}$

STEP 8

Calculate the value of $x$ for the first case:
$x = 5$

STEP 9

Set up the second case where the expression inside the absolute value is negative:
$-(2x - 1) = 9$

STEP 10

Simplify the equation by distributing the negative sign:
$-2x + 1 = 9$

STEP 11

Solve the equation from STEP_10 for $x$ by first subtracting 1 from both sides of the equation:
$-2x = 9 - 1$

STEP 12

Now, simplify the right side of the equation:
$-2x = 8$

STEP 13

Divide both sides of the equation by -2 to solve for $x$ :
$x = \frac{8}{-2}$

STEP 14

Calculate the value of $x$ for the second case:
$x = -4$

STEP 15

Combine the solutions from both cases to provide the complete solution to the absolute value equation:
The solutions to the equation $|2x - 1| = 9$ are $x = 5$ and $x = -4$ .

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Solve the absolute value equation $|2x-1| = 9$ or indicate if it has no solution.

STEP 1

Assumptions
1. We are given the absolute value equation $|2x - 1| = 9$ .
2. We need to find all values of $x$ that satisfy the equation.

STEP 2

Understand that the absolute value of a number is its distance from zero on the number line, which means it is always non-negative. The absolute value equation $|2x - 1| = 9$ tells us that the expression $2x - 1$ is 9 units away from zero on the number line.

STEP 3

Recognize that since $|2x - 1|$ can be either positive or negative inside the absolute value, we need to consider two separate cases to solve for $x$ .

STEP 4

Set up the first case where the expression inside the absolute value is positive:
$2x - 1 = 9$

STEP 5

Solve the equation from STEP_4 for $x$ by first adding 1 to both sides of the equation:
$2x = 9 + 1$

STEP 6

Now, simplify the right side of the equation:
$2x = 10$

STEP 7

Divide both sides of the equation by 2 to solve for $x$ :
$x = \frac{10}{2}$

STEP 8

Calculate the value of $x$ for the first case:
$x = 5$

STEP 9

Set up the second case where the expression inside the absolute value is negative:
$-(2x - 1) = 9$

STEP 10

Simplify the equation by distributing the negative sign:
$-2x + 1 = 9$

STEP 11

Solve the equation from STEP_10 for $x$ by first subtracting 1 from both sides of the equation:
$-2x = 9 - 1$

STEP 12

Now, simplify the right side of the equation:
$-2x = 8$

STEP 13

Divide both sides of the equation by -2 to solve for $x$ :
$x = \frac{8}{-2}$

STEP 14

Calculate the value of $x$ for the second case:
$x = -4$

STEP 15

Combine the solutions from both cases to provide the complete solution to the absolute value equation:
The solutions to the equation $|2x - 1| = 9$ are $x = 5$ and $x = -4$ .

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Solve the absolute value equation ∣2x−1∣=9|2x-1| = 9∣2x−1∣=9 or indicate if it has no solution.

STEP 1

Assumptions1. We are given the absolute value equation ∣2x−1∣=9 |2x - 1| = 9 ∣2x−1∣=9.2. We need to find all values of x x x that satisfy the equation.

STEP 2

Understand that the absolute value of a number is its distance from zero on the number line, which means it is always non-negative. The absolute value equation ∣2x−1∣=9 |2x - 1| = 9 ∣2x−1∣=9 tells us that the expression 2x−1 2x - 1 2x−1 is 9 units away from zero on the number line.

STEP 3

Recognize that since ∣2x−1∣ |2x - 1| ∣2x−1∣ can be either positive or negative inside the absolute value, we need to consider two separate cases to solve for x x x.

STEP 4

Set up the first case where the expression inside the absolute value is positive:2x−1=9 2x - 1 = 9 2x−1=9

STEP 5

Solve the equation from STEP_4 for x x x by first adding 1 to both sides of the equation:2x=9+1 2x = 9 + 1 2x=9+1

STEP 6

Now, simplify the right side of the equation:2x=10 2x = 10 2x=10

STEP 7

Divide both sides of the equation by 2 to solve for x x x:x=102 x = \frac{10}{2} x=210​

STEP 8

Calculate the value of x x x for the first case:x=5 x = 5 x=5

STEP 9

Set up the second case where the expression inside the absolute value is negative:−(2x−1)=9 -(2x - 1) = 9 −(2x−1)=9

STEP 10

Simplify the equation by distributing the negative sign:−2x+1=9 -2x + 1 = 9 −2x+1=9

STEP 11

Solve the equation from STEP_10 for x x x by first subtracting 1 from both sides of the equation:−2x=9−1 -2x = 9 - 1 −2x=9−1

STEP 12

Now, simplify the right side of the equation:−2x=8 -2x = 8 −2x=8

STEP 13

Divide both sides of the equation by -2 to solve for x x x:x=8−2 x = \frac{8}{-2} x=−28​

STEP 14

Calculate the value of x x x for the second case:x=−4 x = -4 x=−4

STEP 15

Combine the solutions from both cases to provide the complete solution to the absolute value equation:The solutions to the equation ∣2x−1∣=9 |2x - 1| = 9 ∣2x−1∣=9 are x=5 x = 5 x=5 and x=−4 x = -4 x=−4.

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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 absolute value equation $|2x-1| = 9$ or indicate if it has no solution.

Assumptions
1. We are given the absolute value equation $|2x - 1| = 9$ .
2. We need to find all values of $x$ that satisfy the equation.

Understand that the absolute value of a number is its distance from zero on the number line, which means it is always non-negative. The absolute value equation $|2x - 1| = 9$ tells us that the expression $2x - 1$ is 9 units away from zero on the number line.

Recognize that since $|2x - 1|$ can be either positive or negative inside the absolute value, we need to consider two separate cases to solve for $x$ .

Set up the first case where the expression inside the absolute value is positive:
$2x - 1 = 9$

Solve the equation from STEP_4 for $x$ by first adding 1 to both sides of the equation:
$2x = 9 + 1$

Now, simplify the right side of the equation:
$2x = 10$

Divide both sides of the equation by 2 to solve for $x$ :
$x = \frac{10}{2}$

Calculate the value of $x$ for the first case:
$x = 5$

Set up the second case where the expression inside the absolute value is negative:
$-(2x - 1) = 9$

Simplify the equation by distributing the negative sign:
$-2x + 1 = 9$

Solve the equation from STEP_10 for $x$ by first subtracting 1 from both sides of the equation:
$-2x = 9 - 1$

Now, simplify the right side of the equation:
$-2x = 8$

Divide both sides of the equation by -2 to solve for $x$ :
$x = \frac{8}{-2}$

Calculate the value of $x$ for the second case:
$x = -4$

Combine the solutions from both cases to provide the complete solution to the absolute value equation:
The solutions to the equation $|2x - 1| = 9$ are $x = 5$ and $x = -4$ .