Math / Algebra

QuestionAbsolute Value Inequalities (Level 2) Score: 1/5 Penalty: 1 off
Question Watch Video Show Examples
Solve the following inequality algebraically. $2|x+10|+4<16$
Answer Attempt 1 out of 2 $\square$ Submit Answer $\phi$ or $\infty$ $<$ $>$ $\leq$ $\geq$ [,] (,) (,) (,) All Real Numbers

Studdy Solution

STEP 1

What is this asking? Find all the values of $x$ that make the expression $2|x+10|+4$ less than 16. Watch out! Don't forget that absolute value expressions represent a distance and can be either positive or zero!

STEP 2

1. Isolate the absolute value
2. Solve for the absolute value
3. Interpret the inequality
4. Find the solution interval

STEP 3

Alright, let's isolate that absolute value!
We want to get $|x+10|$ all by itself on one side of the inequality.
Our inequality is $2|x+10|+4<16$ .

STEP 4

First, let's subtract 4 from both sides.
Why? Because we're trying to undo the addition, getting us closer to isolating the absolute value.
This gives us $2|x+10|+4-4 < 16-4$ , which simplifies to $2|x+10| < 12$ .

STEP 5

Next, let's divide both sides by 2.
Remember, we're trying to completely isolate that absolute value.
This gives us $\frac{2|x+10|}{2} < \frac{12}{2}$ , which simplifies to $|x+10| < 6$ .
Awesome!

STEP 6

Now, we've got $|x+10| < 6$ .
Remember what absolute value means?
It's the distance from zero!
So, this inequality is saying that the distance between $x+10$ and zero is less than 6.

STEP 7

This means $x+10$ can be any number between -6 and 6.
We can write this as two separate inequalities: $x+10 > -6$ and $x+10 < 6$ .
See how we flipped the sign when we introduced the negative?
That's super important!

STEP 8

Let's tackle $x+10 > -6$ first.
We subtract 10 from both sides to isolate $x$ , giving us $x+10-10 > -6-10$ , which simplifies to $x > -16$ .

STEP 9

Now, let's look at $x+10 < 6$ .
Again, we subtract 10 from both sides to isolate $x$ , giving us $x+10-10 < 6-10$ , which simplifies to $x < -4$ .

STEP 10

So, we have $x > -16$ and $x < -4$ .
This means $x$ must be greater than $-16$ and less than $-4$ .
We can write this as a single compound inequality: $-16 < x < -4$ .

STEP 11

This tells us that $x$ can be any number between $-16$ and $-4$ , not including the endpoints.
In interval notation, this is written as $(-16, -4)$ .
Boom!

STEP 12

$(-16, -4)$

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QuestionAbsolute Value Inequalities (Level 2) Score: 1/5 Penalty: 1 off
Question Watch Video Show Examples
Solve the following inequality algebraically. $2|x+10|+4<16$
Answer Attempt 1 out of 2 $\square$ Submit Answer $\phi$ or $\infty$ $<$ $>$ $\leq$ $\geq$ [,] (,) (,) (,) All Real Numbers

Studdy Solution

STEP 1

What is this asking? Find all the values of $x$ that make the expression $2|x+10|+4$ less than 16. Watch out! Don't forget that absolute value expressions represent a distance and can be either positive or zero!

STEP 2

1. Isolate the absolute value
2. Solve for the absolute value
3. Interpret the inequality
4. Find the solution interval

STEP 3

Alright, let's isolate that absolute value!
We want to get $|x+10|$ all by itself on one side of the inequality.
Our inequality is $2|x+10|+4<16$ .

STEP 4

First, let's subtract 4 from both sides.
Why? Because we're trying to undo the addition, getting us closer to isolating the absolute value.
This gives us $2|x+10|+4-4 < 16-4$ , which simplifies to $2|x+10| < 12$ .

STEP 5

Next, let's divide both sides by 2.
Remember, we're trying to completely isolate that absolute value.
This gives us $\frac{2|x+10|}{2} < \frac{12}{2}$ , which simplifies to $|x+10| < 6$ .
Awesome!

STEP 6

Now, we've got $|x+10| < 6$ .
Remember what absolute value means?
It's the distance from zero!
So, this inequality is saying that the distance between $x+10$ and zero is less than 6.

STEP 7

This means $x+10$ can be any number between -6 and 6.
We can write this as two separate inequalities: $x+10 > -6$ and $x+10 < 6$ .
See how we flipped the sign when we introduced the negative?
That's super important!

STEP 8

Let's tackle $x+10 > -6$ first.
We subtract 10 from both sides to isolate $x$ , giving us $x+10-10 > -6-10$ , which simplifies to $x > -16$ .

STEP 9

Now, let's look at $x+10 < 6$ .
Again, we subtract 10 from both sides to isolate $x$ , giving us $x+10-10 < 6-10$ , which simplifies to $x < -4$ .

STEP 10

So, we have $x > -16$ and $x < -4$ .
This means $x$ must be greater than $-16$ and less than $-4$ .
We can write this as a single compound inequality: $-16 < x < -4$ .

STEP 11

This tells us that $x$ can be any number between $-16$ and $-4$ , not including the endpoints.
In interval notation, this is written as $(-16, -4)$ .
Boom!

STEP 12

$(-16, -4)$

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Studdy Solution

STEP 1

What is this asking? Find all the values of xxx that make the expression 2∣x+10∣+42|x+10|+42∣x+10∣+4 less than 16. Watch out! Don't forget that absolute value expressions represent a *distance* and can be either positive or zero!

STEP 2

1. Isolate the absolute value2. Solve for the absolute value3. Interpret the inequality4. Find the solution interval

STEP 3

Alright, let's **isolate** that absolute value!We want to get ∣x+10∣|x+10|∣x+10∣ all by itself on one side of the inequality.Our inequality is 2∣x+10∣+4<162|x+10|+4<162∣x+10∣+4<16.

STEP 4

First, let's **subtract** 4 from both sides.Why? Because we're trying to *undo* the addition, getting us closer to isolating the absolute value.This gives us 2∣x+10∣+4−4<16−42|x+10|+4-4 < 16-42∣x+10∣+4−4<16−4, which simplifies to 2∣x+10∣<122|x+10| < 122∣x+10∣<12.

STEP 5

Next, let's **divide** both sides by 2.Remember, we're trying to completely isolate that absolute value.This gives us 2∣x+10∣2<122\frac{2|x+10|}{2} < \frac{12}{2}22∣x+10∣​<212​, which simplifies to ∣x+10∣<6|x+10| < 6∣x+10∣<6.Awesome!

STEP 6

Now, we've got ∣x+10∣<6|x+10| < 6∣x+10∣<6.Remember what absolute value means?It's the *distance* from zero!So, this inequality is saying that the distance between x+10x+10x+10 and zero is less than 6.

STEP 7

This means x+10x+10x+10 can be any number between -6 and 6.We can write this as two separate inequalities: x+10>−6x+10 > -6x+10>−6 and x+10<6x+10 < 6x+10<6.See how we flipped the sign when we introduced the negative?That's super important!

STEP 8

Let's tackle x+10>−6x+10 > -6x+10>−6 first.We **subtract** 10 from both sides to isolate xxx, giving us x+10−10>−6−10x+10-10 > -6-10x+10−10>−6−10, which simplifies to x>−16x > -16x>−16.

STEP 9

Now, let's look at x+10<6x+10 < 6x+10<6.Again, we **subtract** 10 from both sides to isolate xxx, giving us x+10−10<6−10x+10-10 < 6-10x+10−10<6−10, which simplifies to x<−4x < -4x<−4.

STEP 10

So, we have x>−16x > -16x>−16 and x<−4x < -4x<−4.This means xxx must be greater than −16-16−16 *and* less than −4-4−4.We can write this as a single compound inequality: −16<x<−4-16 < x < -4−16<x<−4.

STEP 11

This tells us that xxx can be any number between −16-16−16 and −4-4−4, not including the endpoints.In interval notation, this is written as (−16,−4)(-16, -4)(−16,−4).Boom!

STEP 12

(−16,−4)(-16, -4)(−16,−4)

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What is this asking? Find all the values of $x$ that make the expression $2|x+10|+4$ less than 16. Watch out! Don't forget that absolute value expressions represent a distance and can be either positive or zero!

1. Isolate the absolute value
2. Solve for the absolute value
3. Interpret the inequality
4. Find the solution interval

Alright, let's isolate that absolute value!
We want to get $|x+10|$ all by itself on one side of the inequality.
Our inequality is $2|x+10|+4<16$ .

First, let's subtract 4 from both sides.
Why? Because we're trying to undo the addition, getting us closer to isolating the absolute value.
This gives us $2|x+10|+4-4 < 16-4$ , which simplifies to $2|x+10| < 12$ .

Next, let's divide both sides by 2.
Remember, we're trying to completely isolate that absolute value.
This gives us $\frac{2|x+10|}{2} < \frac{12}{2}$ , which simplifies to $|x+10| < 6$ .
Awesome!

Now, we've got $|x+10| < 6$ .
Remember what absolute value means?
It's the distance from zero!
So, this inequality is saying that the distance between $x+10$ and zero is less than 6.

This means $x+10$ can be any number between -6 and 6.
We can write this as two separate inequalities: $x+10 > -6$ and $x+10 < 6$ .
See how we flipped the sign when we introduced the negative?
That's super important!

Let's tackle $x+10 > -6$ first.
We subtract 10 from both sides to isolate $x$ , giving us $x+10-10 > -6-10$ , which simplifies to $x > -16$ .

Now, let's look at $x+10 < 6$ .
Again, we subtract 10 from both sides to isolate $x$ , giving us $x+10-10 < 6-10$ , which simplifies to $x < -4$ .

So, we have $x > -16$ and $x < -4$ .
This means $x$ must be greater than $-16$ and less than $-4$ .
We can write this as a single compound inequality: $-16 < x < -4$ .

This tells us that $x$ can be any number between $-16$ and $-4$ , not including the endpoints.
In interval notation, this is written as $(-16, -4)$ .
Boom!

$(-16, -4)$