Solved on Jan 04, 2024

Find the values of $x$ that satisfy $|x-3|=6$ .

STEP 1

Assumptions
1. We are given the absolute value equation $|x-3|=6$ .
2. We need to find all values of $x$ that satisfy the equation.

STEP 2

The absolute value equation $|x-3|=6$ means that the expression inside the absolute value, $x-3$ , can either be $6$ or $-6$ because the absolute value of a number is the distance from zero on the number line, which is always non-negative.

STEP 3

We will create two separate equations to solve for $x$ based on the two possible cases for the absolute value:
1. $x - 3 = 6$
2. $x - 3 = -6$

STEP 4

First, we solve the equation $x - 3 = 6$ for $x$ .
$x - 3 = 6$

STEP 5

Add $3$ to both sides of the equation to isolate $x$ .
$x = 6 + 3$

STEP 6

Calculate the value of $x$ for the first case.
$x = 9$

STEP 7

Now, we solve the second equation $x - 3 = -6$ for $x$ .
$x - 3 = -6$

STEP 8

Add $3$ to both sides of the equation to isolate $x$ .
$x = -6 + 3$

STEP 9

Calculate the value of $x$ for the second case.
$x = -3$

STEP 10

Combine the solutions from both cases to express all values of $x$ that satisfy the original absolute value equation.
The solutions are $x = 9$ and $x = -3$ .

STEP 11

Match the solutions to the given options.
The correct answer is (A) $x=-3, x=9$ .

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Find the values of $x$ that satisfy $|x-3|=6$ .

STEP 1

Assumptions
1. We are given the absolute value equation $|x-3|=6$ .
2. We need to find all values of $x$ that satisfy the equation.

STEP 2

The absolute value equation $|x-3|=6$ means that the expression inside the absolute value, $x-3$ , can either be $6$ or $-6$ because the absolute value of a number is the distance from zero on the number line, which is always non-negative.

STEP 3

We will create two separate equations to solve for $x$ based on the two possible cases for the absolute value:
1. $x - 3 = 6$
2. $x - 3 = -6$

STEP 4

First, we solve the equation $x - 3 = 6$ for $x$ .
$x - 3 = 6$

STEP 5

Add $3$ to both sides of the equation to isolate $x$ .
$x = 6 + 3$

STEP 6

Calculate the value of $x$ for the first case.
$x = 9$

STEP 7

Now, we solve the second equation $x - 3 = -6$ for $x$ .
$x - 3 = -6$

STEP 8

Add $3$ to both sides of the equation to isolate $x$ .
$x = -6 + 3$

STEP 9

Calculate the value of $x$ for the second case.
$x = -3$

STEP 10

Combine the solutions from both cases to express all values of $x$ that satisfy the original absolute value equation.
The solutions are $x = 9$ and $x = -3$ .

STEP 11

Match the solutions to the given options.
The correct answer is (A) $x=-3, x=9$ .

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Find the values of xxx that satisfy ∣x−3∣=6|x-3|=6∣x−3∣=6.

STEP 1

Assumptions1. We are given the absolute value equation ∣x−3∣=6|x-3|=6∣x−3∣=6.2. We need to find all values of xxx that satisfy the equation.

STEP 2

The absolute value equation ∣x−3∣=6|x-3|=6∣x−3∣=6 means that the expression inside the absolute value, x−3x-3x−3, can either be 666 or −6-6−6 because the absolute value of a number is the distance from zero on the number line, which is always non-negative.

STEP 3

We will create two separate equations to solve for xxx based on the two possible cases for the absolute value:1. x−3=6x - 3 = 6x−3=62. x−3=−6x - 3 = -6x−3=−6

STEP 4

First, we solve the equation x−3=6x - 3 = 6x−3=6 for xxx.x−3=6x - 3 = 6x−3=6

STEP 5

Add 333 to both sides of the equation to isolate xxx.x=6+3x = 6 + 3x=6+3

STEP 6

Calculate the value of xxx for the first case.x=9x = 9x=9

STEP 7

Now, we solve the second equation x−3=−6x - 3 = -6x−3=−6 for xxx.x−3=−6x - 3 = -6x−3=−6

STEP 8

Add 333 to both sides of the equation to isolate xxx.x=−6+3x = -6 + 3x=−6+3

STEP 9

Calculate the value of xxx for the second case.x=−3x = -3x=−3

STEP 10

Combine the solutions from both cases to express all values of xxx that satisfy the original absolute value equation.The solutions are x=9x = 9x=9 and x=−3x = -3x=−3.

STEP 11

Match the solutions to the given options.The correct answer is (A) x=−3,x=9x=-3, x=9x=−3,x=9.

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Find the values of $x$ that satisfy $|x-3|=6$ .

Assumptions
1. We are given the absolute value equation $|x-3|=6$ .
2. We need to find all values of $x$ that satisfy the equation.

The absolute value equation $|x-3|=6$ means that the expression inside the absolute value, $x-3$ , can either be $6$ or $-6$ because the absolute value of a number is the distance from zero on the number line, which is always non-negative.

We will create two separate equations to solve for $x$ based on the two possible cases for the absolute value:
1. $x - 3 = 6$
2. $x - 3 = -6$

First, we solve the equation $x - 3 = 6$ for $x$ .
$x - 3 = 6$

Add $3$ to both sides of the equation to isolate $x$ .
$x = 6 + 3$

Calculate the value of $x$ for the first case.
$x = 9$

Now, we solve the second equation $x - 3 = -6$ for $x$ .
$x - 3 = -6$

Add $3$ to both sides of the equation to isolate $x$ .
$x = -6 + 3$

Calculate the value of $x$ for the second case.
$x = -3$

Combine the solutions from both cases to express all values of $x$ that satisfy the original absolute value equation.
The solutions are $x = 9$ and $x = -3$ .

Match the solutions to the given options.
The correct answer is (A) $x=-3, x=9$ .