Math / Algebra

QuestionSolve the following absolute value inequality. $\begin{array}{l} \frac{4|x-9|}{3}>8 \\ x>[?] \\ x<\square \square \end{array}$ Enter

Studdy Solution

STEP 1

What is this asking? We need to find the values of $x$ that make the inequality $\frac{4|x-9|}{3} > 8$ true.
Basically, we're figuring out how far $x$ can stray from 9 before the expression gets too big! Watch out! Don't forget how absolute values work!
They always make things positive, so we need to consider both cases where $x-9$ is positive and where it's negative.

STEP 2

1. Isolate the Absolute Value
2. Consider Positive Case
3. Consider Negative Case
4. Combine Solutions

STEP 3

To get the absolute value by itself, let's multiply both sides of the inequality by $\frac{3}{4}$ .
We're doing this to divide to one on the left side, since $\frac{3}{4} \cdot \frac{4}{3} = 1$ .
This gives us: $\frac{3}{4} \cdot \frac{4|x-9|}{3} > \frac{3}{4} \cdot 8$

STEP 4

This simplifies to: $|x-9| > 6$ Now we're ready to tackle those absolute values!

STEP 5

If $x-9$ is positive or zero, then $|x-9|$ is just $x-9$ .
So, we have: $x-9 > 6$

STEP 6

Add 9 to both sides (to add to zero on the left side) to get: $x > 15$ So, any $x$ greater than 15 works!

STEP 7

If $x-9$ is negative, then $|x-9|$ is $-(x-9)$ , which is the same as $9-x$ .
So, we have: $9-x > 6$

STEP 8

Subtract 9 from both sides (to add to zero on the left side): $-x > -3$

STEP 9

Now, multiply both sides by -1 (to multiply to one on the left side).
Remember to flip the inequality sign when multiplying by a negative number! $x < 3$ So, any $x$ less than 3 also works!

STEP 10

We found that $x > 15$ or $x < 3$ .
That's our solution!

STEP 11

$x > 15$ $x < 3$

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QuestionSolve the following absolute value inequality. $\begin{array}{l} \frac{4|x-9|}{3}>8 \\ x>[?] \\ x<\square \square \end{array}$ Enter

Studdy Solution

STEP 1

STEP 2

1. Isolate the Absolute Value
2. Consider Positive Case
3. Consider Negative Case
4. Combine Solutions

STEP 3

To get the absolute value by itself, let's multiply both sides of the inequality by $\frac{3}{4}$ .
We're doing this to divide to one on the left side, since $\frac{3}{4} \cdot \frac{4}{3} = 1$ .
This gives us: $\frac{3}{4} \cdot \frac{4|x-9|}{3} > \frac{3}{4} \cdot 8$

STEP 4

This simplifies to: $|x-9| > 6$ Now we're ready to tackle those absolute values!

STEP 5

If $x-9$ is positive or zero, then $|x-9|$ is just $x-9$ .
So, we have: $x-9 > 6$

STEP 6

Add 9 to both sides (to add to zero on the left side) to get: $x > 15$ So, any $x$ greater than 15 works!

STEP 7

If $x-9$ is negative, then $|x-9|$ is $-(x-9)$ , which is the same as $9-x$ .
So, we have: $9-x > 6$

STEP 8

Subtract 9 from both sides (to add to zero on the left side): $-x > -3$

STEP 9

Now, multiply both sides by -1 (to multiply to one on the left side).
Remember to flip the inequality sign when multiplying by a negative number! $x < 3$ So, any $x$ less than 3 also works!

STEP 10

We found that $x > 15$ or $x < 3$ .
That's our solution!

STEP 11

$x > 15$ $x < 3$

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QuestionSolve the following absolute value inequality. 4∣x−9∣3>8x>[?]x<□□\begin{array}{l} \frac{4|x-9|}{3}>8 \\ x>[?] \\ x<\square \square \end{array}34∣x−9∣​>8x>[?]x<□□​ Enter

Studdy Solution

STEP 1

STEP 2

1. Isolate the Absolute Value2. Consider Positive Case3. Consider Negative Case4. Combine Solutions

STEP 3

STEP 4

This simplifies to: ∣x−9∣>6 |x-9| > 6 ∣x−9∣>6 Now we're ready to tackle those absolute values!

STEP 5

If x−9x-9x−9 is positive or zero, then ∣x−9∣|x-9|∣x−9∣ is just x−9x-9x−9.So, we have: x−9>6 x-9 > 6 x−9>6

STEP 6

**Add** 9 to both sides (to add to zero on the left side) to get: x>15 x > 15 x>15 So, any xxx greater than 15 works!

STEP 7

If x−9x-9x−9 is negative, then ∣x−9∣|x-9|∣x−9∣ is −(x−9)-(x-9)−(x−9), which is the same as 9−x9-x9−x.So, we have: 9−x>6 9-x > 6 9−x>6

STEP 8

**Subtract** 9 from both sides (to add to zero on the left side): −x>−3 -x > -3 −x>−3

STEP 9

Now, **multiply** both sides by -1 (to multiply to one on the left side).Remember to flip the inequality sign when multiplying by a negative number! x<3 x < 3 x<3 So, any xxx less than 3 also works!

STEP 10

We found that x>15x > 15x>15 or x<3x < 3x<3.That's our solution!

STEP 11

x>15x > 15x>15 x<3x < 3x<3

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QuestionSolve the following absolute value inequality. $\begin{array}{l} \frac{4|x-9|}{3}>8 \\ x>[?] \\ x<\square \square \end{array}$ Enter

1. Isolate the Absolute Value
2. Consider Positive Case
3. Consider Negative Case
4. Combine Solutions

This simplifies to: $|x-9| > 6$ Now we're ready to tackle those absolute values!

If $x-9$ is positive or zero, then $|x-9|$ is just $x-9$ .
So, we have: $x-9 > 6$

Add 9 to both sides (to add to zero on the left side) to get: $x > 15$ So, any $x$ greater than 15 works!

If $x-9$ is negative, then $|x-9|$ is $-(x-9)$ , which is the same as $9-x$ .
So, we have: $9-x > 6$

Subtract 9 from both sides (to add to zero on the left side): $-x > -3$

Now, multiply both sides by -1 (to multiply to one on the left side).
Remember to flip the inequality sign when multiplying by a negative number! $x < 3$ So, any $x$ less than 3 also works!

We found that $x > 15$ or $x < 3$ .
That's our solution!

$x > 15$ $x < 3$