Question Find all numbers $x$ that are 9 units away from 11, expressed using absolute value.

Studdy Solution

STEP 1

Assumptions
1. The distance between two numbers on the number line is the absolute value of their difference.
2. The number from which the distance is measured is 11.
3. The distance we are interested in is 9.

STEP 2

To describe the numbers that are at a distance of 9 from the number 11, we use the concept of absolute value. The absolute value of a number is its distance from 0 on the number line.
$|x - a| = b$
Here, $a$ is the number from which we are measuring the distance, $b$ is the distance, and $x$ is the number we are trying to find.

STEP 3

In our case, $a = 11$ and $b = 9$ . We can write the equation as:
$|x - 11| = 9$

STEP 4

The absolute value equation $|x - 11| = 9$ means that the distance between $x$ and 11 is 9. This can happen in two cases:
1. $x$ is 9 units greater than 11.
2. $x$ is 9 units less than 11.

STEP 5

For the first case, where $x$ is 9 units greater than 11, we can write:
$x - 11 = 9$

STEP 6

Solve for $x$ in the first case:
$x = 9 + 11$
$x = 20$

STEP 7

For the second case, where $x$ is 9 units less than 11, we can write:
$x - 11 = -9$

STEP 8

Solve for $x$ in the second case:
$x = -9 + 11$
$x = 2$

STEP 9

Combining both cases, the numbers that are at a distance of 9 from the number 11 are 2 and 20. We can express this using absolute value notation as:
$|x - 11| = 9$
The solutions are $x = 2$ and $x = 20$ .

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Question Find all numbers $x$ that are 9 units away from 11, expressed using absolute value.

Studdy Solution

STEP 1

Assumptions
1. The distance between two numbers on the number line is the absolute value of their difference.
2. The number from which the distance is measured is 11.
3. The distance we are interested in is 9.

STEP 2

STEP 3

In our case, $a = 11$ and $b = 9$ . We can write the equation as:
$|x - 11| = 9$

STEP 4

The absolute value equation $|x - 11| = 9$ means that the distance between $x$ and 11 is 9. This can happen in two cases:
1. $x$ is 9 units greater than 11.
2. $x$ is 9 units less than 11.

STEP 5

For the first case, where $x$ is 9 units greater than 11, we can write:
$x - 11 = 9$

STEP 6

Solve for $x$ in the first case:
$x = 9 + 11$
$x = 20$

STEP 7

For the second case, where $x$ is 9 units less than 11, we can write:
$x - 11 = -9$

STEP 8

Solve for $x$ in the second case:
$x = -9 + 11$
$x = 2$

STEP 9

Combining both cases, the numbers that are at a distance of 9 from the number 11 are 2 and 20. We can express this using absolute value notation as:
$|x - 11| = 9$
The solutions are $x = 2$ and $x = 20$ .

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Question Find all numbers xxx that are 9 units away from 11, expressed using absolute value.

Studdy Solution

STEP 1

Assumptions1. The distance between two numbers on the number line is the absolute value of their difference.2. The number from which the distance is measured is 11.3. The distance we are interested in is 9.

STEP 2

STEP 3

In our case, a=11a = 11a=11 and b=9b = 9b=9. We can write the equation as:∣x−11∣=9|x - 11| = 9∣x−11∣=9

STEP 4

The absolute value equation ∣x−11∣=9|x - 11| = 9∣x−11∣=9 means that the distance between xxx and 11 is 9. This can happen in two cases:1. xxx is 9 units greater than 11.2. xxx is 9 units less than 11.

STEP 5

For the first case, where xxx is 9 units greater than 11, we can write:x−11=9x - 11 = 9x−11=9

STEP 6

Solve for xxx in the first case:x=9+11x = 9 + 11x=9+11x=20x = 20x=20

STEP 7

For the second case, where xxx is 9 units less than 11, we can write:x−11=−9x - 11 = -9x−11=−9

STEP 8

Solve for xxx in the second case:x=−9+11x = -9 + 11x=−9+11x=2x = 2x=2

STEP 9

Combining both cases, the numbers that are at a distance of 9 from the number 11 are 2 and 20. We can express this using absolute value notation as:∣x−11∣=9|x - 11| = 9∣x−11∣=9The solutions are x=2x = 2x=2 and x=20x = 20x=20.

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Question Find all numbers $x$ that are 9 units away from 11, expressed using absolute value.

Assumptions
1. The distance between two numbers on the number line is the absolute value of their difference.
2. The number from which the distance is measured is 11.
3. The distance we are interested in is 9.

In our case, $a = 11$ and $b = 9$ . We can write the equation as:
$|x - 11| = 9$

The absolute value equation $|x - 11| = 9$ means that the distance between $x$ and 11 is 9. This can happen in two cases:
1. $x$ is 9 units greater than 11.
2. $x$ is 9 units less than 11.

For the first case, where $x$ is 9 units greater than 11, we can write:
$x - 11 = 9$

Solve for $x$ in the first case:
$x = 9 + 11$
$x = 20$

For the second case, where $x$ is 9 units less than 11, we can write:
$x - 11 = -9$

Solve for $x$ in the second case:
$x = -9 + 11$
$x = 2$

Combining both cases, the numbers that are at a distance of 9 from the number 11 are 2 and 20. We can express this using absolute value notation as:
$|x - 11| = 9$
The solutions are $x = 2$ and $x = 20$ .