Solved on Jan 21, 2024

Solve the inequality $2|x-8|+4<13$ , and express the solution in the form $x<A$ or $x>B$ , where $A<x<B$ .

STEP 1

Assumptions
1. We are solving the inequality $2|x-8|+4<13$ .
2. We will consider two cases for the absolute value: one where $x-8$ is positive or zero, and another where $x-8$ is negative.
3. We will solve each case separately and then combine the solutions to find the final answer.

STEP 2

First, we isolate the absolute value term on one side of the inequality.
$2|x-8| < 13 - 4$

STEP 3

Simplify the right side of the inequality.
$2|x-8| < 9$

STEP 4

Now, we divide both sides of the inequality by 2 to solve for the absolute value.
$|x-8| < \frac{9}{2}$

STEP 5

Convert the fraction to a decimal to make it easier to interpret.
$|x-8| < 4.5$

STEP 6

Now, we consider the two cases for the absolute value.
Case 1: $x-8$ is positive or zero, which means $x-8 \geq 0$ .
Case 2: $x-8$ is negative, which means $x-8 < 0$ .

STEP 7

Solve Case 1:
If $x-8 \geq 0$ , then $|x-8| = x-8$ .
So we have $x-8 < 4.5$ .

STEP 8

Add 8 to both sides of the inequality to solve for $x$ in Case 1.
$x < 4.5 + 8$

STEP 9

Calculate the value of $x$ for Case 1.
$x < 12.5$

STEP 10

Now, solve Case 2:
If $x-8 < 0$ , then $|x-8| = -(x-8)$ .
So we have $-(x-8) < 4.5$ .

STEP 11

Distribute the negative sign on the left side of the inequality.
$-x+8 < 4.5$

STEP 12

Add $x$ to both sides of the inequality to move $x$ to the right side.
$8 < x + 4.5$

STEP 13

Subtract 4.5 from both sides of the inequality to solve for $x$ in Case 2.
$8 - 4.5 < x$

STEP 14

Calculate the value of $x$ for Case 2.
$3.5 < x$

STEP 15

Combine the solutions from Case 1 and Case 2.
From Case 1, we have $x < 12.5$ .
From Case 2, we have $3.5 < x$ .

STEP 16

Write the final solution by combining the two inequalities.
$3.5 < x < 12.5$
This means that $x$ must be greater than 3.5 and less than 12.5.
The answer is in the form $\mathrm{A}<\mathrm{x}<\mathrm{B}$ where $\mathrm{A}=3.5$ and $\mathrm{B}=12.5$ .

Was this helpful?

Solve the inequality $2|x-8|+4<13$ , and express the solution in the form $x<A$ or $x>B$ , where $A<x<B$ .

STEP 1

Assumptions
1. We are solving the inequality $2|x-8|+4<13$ .
2. We will consider two cases for the absolute value: one where $x-8$ is positive or zero, and another where $x-8$ is negative.
3. We will solve each case separately and then combine the solutions to find the final answer.

STEP 2

First, we isolate the absolute value term on one side of the inequality.
$2|x-8| < 13 - 4$

STEP 3

Simplify the right side of the inequality.
$2|x-8| < 9$

STEP 4

Now, we divide both sides of the inequality by 2 to solve for the absolute value.
$|x-8| < \frac{9}{2}$

STEP 5

Convert the fraction to a decimal to make it easier to interpret.
$|x-8| < 4.5$

STEP 6

Now, we consider the two cases for the absolute value.
Case 1: $x-8$ is positive or zero, which means $x-8 \geq 0$ .
Case 2: $x-8$ is negative, which means $x-8 < 0$ .

STEP 7

Solve Case 1:
If $x-8 \geq 0$ , then $|x-8| = x-8$ .
So we have $x-8 < 4.5$ .

STEP 8

Add 8 to both sides of the inequality to solve for $x$ in Case 1.
$x < 4.5 + 8$

STEP 9

Calculate the value of $x$ for Case 1.
$x < 12.5$

STEP 10

Now, solve Case 2:
If $x-8 < 0$ , then $|x-8| = -(x-8)$ .
So we have $-(x-8) < 4.5$ .

STEP 11

Distribute the negative sign on the left side of the inequality.
$-x+8 < 4.5$

STEP 12

Add $x$ to both sides of the inequality to move $x$ to the right side.
$8 < x + 4.5$

STEP 13

Subtract 4.5 from both sides of the inequality to solve for $x$ in Case 2.
$8 - 4.5 < x$

STEP 14

Calculate the value of $x$ for Case 2.
$3.5 < x$

STEP 15

Combine the solutions from Case 1 and Case 2.
From Case 1, we have $x < 12.5$ .
From Case 2, we have $3.5 < x$ .

STEP 16

Write the final solution by combining the two inequalities.
$3.5 < x < 12.5$
This means that $x$ must be greater than 3.5 and less than 12.5.
The answer is in the form $\mathrm{A}<\mathrm{x}<\mathrm{B}$ where $\mathrm{A}=3.5$ and $\mathrm{B}=12.5$ .

Start learning now

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

Solve the inequality 2∣x−8∣+4<132|x-8|+4<132∣x−8∣+4<13, and express the solution in the form x<Ax<Ax<A or x>Bx>Bx>B, where A<x<BA<x<BA<x<B.

STEP 1

STEP 2

First, we isolate the absolute value term on one side of the inequality.2∣x−8∣<13−42|x-8| < 13 - 42∣x−8∣<13−4

STEP 3

Simplify the right side of the inequality.2∣x−8∣<92|x-8| < 92∣x−8∣<9

STEP 4

Now, we divide both sides of the inequality by 2 to solve for the absolute value.∣x−8∣<92|x-8| < \frac{9}{2}∣x−8∣<29​

STEP 5

Convert the fraction to a decimal to make it easier to interpret.∣x−8∣<4.5|x-8| < 4.5∣x−8∣<4.5

STEP 6

Now, we consider the two cases for the absolute value.Case 1: x−8x-8x−8 is positive or zero, which means x−8≥0x-8 \geq 0x−8≥0.Case 2: x−8x-8x−8 is negative, which means x−8<0x-8 < 0x−8<0.

STEP 7

Solve Case 1:If x−8≥0x-8 \geq 0x−8≥0, then ∣x−8∣=x−8|x-8| = x-8∣x−8∣=x−8.So we have x−8<4.5x-8 < 4.5x−8<4.5.

STEP 8

Add 8 to both sides of the inequality to solve for xxx in Case 1.x<4.5+8x < 4.5 + 8x<4.5+8

STEP 9

Calculate the value of xxx for Case 1.x<12.5x < 12.5x<12.5

STEP 10

Now, solve Case 2:If x−8<0x-8 < 0x−8<0, then ∣x−8∣=−(x−8)|x-8| = -(x-8)∣x−8∣=−(x−8).So we have −(x−8)<4.5-(x-8) < 4.5−(x−8)<4.5.

STEP 11

Distribute the negative sign on the left side of the inequality.−x+8<4.5-x+8 < 4.5−x+8<4.5

STEP 12

Add xxx to both sides of the inequality to move xxx to the right side.8<x+4.58 < x + 4.58<x+4.5

STEP 13

Subtract 4.5 from both sides of the inequality to solve for xxx in Case 2.8−4.5<x8 - 4.5 < x8−4.5<x

STEP 14

Calculate the value of xxx for Case 2.3.5<x3.5 < x3.5<x

STEP 15

Combine the solutions from Case 1 and Case 2.From Case 1, we have x<12.5x < 12.5x<12.5.From Case 2, we have 3.5<x3.5 < x3.5<x.

STEP 16

Write the final solution by combining the two inequalities.3.5<x<12.53.5 < x < 12.53.5<x<12.5This means that xxx must be greater than 3.5 and less than 12.5.The answer is in the form A<x<B\mathrm{A}<\mathrm{x}<\mathrm{B}A<x<B where A=3.5\mathrm{A}=3.5A=3.5 and B=12.5\mathrm{B}=12.5B=12.5.

Start learning now

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

Solve the inequality $2|x-8|+4<13$ , and express the solution in the form $x<A$ or $x>B$ , where $A<x<B$ .

First, we isolate the absolute value term on one side of the inequality.
$2|x-8| < 13 - 4$

Simplify the right side of the inequality.
$2|x-8| < 9$

Now, we divide both sides of the inequality by 2 to solve for the absolute value.
$|x-8| < \frac{9}{2}$

Convert the fraction to a decimal to make it easier to interpret.
$|x-8| < 4.5$

Now, we consider the two cases for the absolute value.
Case 1: $x-8$ is positive or zero, which means $x-8 \geq 0$ .
Case 2: $x-8$ is negative, which means $x-8 < 0$ .

Solve Case 1:
If $x-8 \geq 0$ , then $|x-8| = x-8$ .
So we have $x-8 < 4.5$ .

Add 8 to both sides of the inequality to solve for $x$ in Case 1.
$x < 4.5 + 8$

Calculate the value of $x$ for Case 1.
$x < 12.5$

Now, solve Case 2:
If $x-8 < 0$ , then $|x-8| = -(x-8)$ .
So we have $-(x-8) < 4.5$ .

Distribute the negative sign on the left side of the inequality.
$-x+8 < 4.5$

Add $x$ to both sides of the inequality to move $x$ to the right side.
$8 < x + 4.5$

Subtract 4.5 from both sides of the inequality to solve for $x$ in Case 2.
$8 - 4.5 < x$

Calculate the value of $x$ for Case 2.
$3.5 < x$

Combine the solutions from Case 1 and Case 2.
From Case 1, we have $x < 12.5$ .
From Case 2, we have $3.5 < x$ .

Write the final solution by combining the two inequalities.
$3.5 < x < 12.5$
This means that $x$ must be greater than 3.5 and less than 12.5.
The answer is in the form $\mathrm{A}<\mathrm{x}<\mathrm{B}$ where $\mathrm{A}=3.5$ and $\mathrm{B}=12.5$ .