I'm working on Lesson 6: Absolute Value Equations and Inequalities Algebra 2 A Unit 2: Expressions, Equations, and Inequalities and feel confident in my first few answers, but beyond that am lost. May I have some help?

**=the answer I believe to be true

1.)
The sides of a triangle are in the ratio 3:4:5. If the triangles perimeter 90cm, what is the length of each side?
a.)22.5 cm, 30 cm, 37.5**
b.)19.3, 25.7, 32.1
c.)7.5, 11.5, 32.1
d.)10.5, 11.5, 12.5

2.)
Is the following sometimes, always, or never true?

6+8x-9=11x+14-3x

I think it's never c.)

3.)
Solve the inequality

r is less than or equal to one fifth

I think the answer is a.)

4.)
Is the following sometimes always or never true?

2(8x-4)-2x <= 14x +12

I think the answer is always true

5.)
Solve the compound inequality

6x-4<-34 or 3x +10>10

I think the answer is x<-5 or x>0 b.)

6.)
-2 <= 2x-4 < 4

I think the answer is 1<= x < 4 c.)

7.) Solve the absolute value equation

2|3x-5|-8=8

I believe the answer is x=-1 or x=4 1/3 b.)

From here on out I am unsure or the answers. I imagine anybody looking at the questions is from Connections academy. If you would be so willing as to help me out I would greatly appreciate it. Have a lovely day.

#3, you list no choices, but the question itself is your answer:

r ≤ 1/5

the rest are correct up to #7

1.) You got it right! The correct answer is a.) 22.5 cm, 30 cm, 37.5.

2.) You're correct again! The equation is never true, so the answer is c.) never.

3.) Correct! The answer is a.) r <= 1/5.

4.) Oops, sorry! The answer is sometimes true. So the correct answer is a.) sometimes.

5.) Nicely done! The answer is indeed b.) x < -5 or x > 0.

6.) Almost there! The correct answer is actually b.) -2 <= x < 4.

7.) Oops, sorry again! The correct answer is c.) x = 5/3 or x = 7/3.

Keep up the good work! If you have any more questions, feel free to ask. Have a great day!

I'd be happy to help you with your questions. Let's go through each one step-by-step:

1.) The sides of a triangle are in the ratio 3:4:5. If the triangle's perimeter is 90cm, what is the length of each side?

To solve this problem, you can set up the equation:

3x + 4x + 5x = 90

Combining like terms, we get:

12x = 90

Dividing both sides by 12, we get:

x = 7.5

Now, we can substitute x back into the original expressions to find the length of each side:

Side 1 = 3x = 3(7.5) = 22.5 cm
Side 2 = 4x = 4(7.5) = 30 cm
Side 3 = 5x = 5(7.5) = 37.5 cm

So, the correct answer is a.) 22.5 cm, 30 cm, 37.5

2.) Is the following sometimes, always, or never true?
6 + 8x - 9 = 11x + 14 - 3x

To solve this equation, we can simplify both sides by combining like terms:

6 - 9 + 14 = 11x - 3x - 8x

11 = 0

The equation 11 = 0 is false. Therefore, the correct answer is never true, which is c.)

3.) Solve the inequality: r is less than or equal to one fifth
The inequality can be written as:
r <= 1/5

So, the correct answer is a.) r <= 1/5

4.) Is the following sometimes, always, or never true?
2(8x - 4) - 2x <= 14x + 12

To solve this inequality, we can simplify both sides by expanding, distributing, and combining like terms:

16x - 8 - 2x <= 14x + 12

14x - 8 <= 14x + 12

The expression 14x - 8 is always less than or equal to 14x + 12. Therefore, the correct answer is always true.

5.) Solve the compound inequality:
6x - 4 < -34 or 3x + 10 > 10

To solve this compound inequality, we solve each inequality separately and then combine the solutions:

6x - 4 < -34

Adding 4 to both sides:

6x < -30

Dividing both sides by 6:

x < -5

Now, solving the second inequality:

3x + 10 > 10

Subtracting 10 from both sides:

3x > 0

Dividing both sides by 3:

x > 0

Combining the solutions, we have:

x < -5 or x > 0

So, the correct answer is b.) x < -5 or x > 0

6.) -2 <= 2x - 4 < 4

To solve this inequality, we need to isolate x. We can do this by adding 4 to all parts of the inequality:

-2 + 4 <= 2x - 4 + 4 < 4 + 4

2 <= 2x < 8

Now, divide all parts by 2:

2/2 <= 2x/2 < 8/2

1 <= x < 4

So, the correct answer is c.) 1 <= x < 4

7.) Solve the absolute value equation:
2|3x - 5| - 8 = 8

To solve this equation, we can isolate the absolute value term first by adding 8 to both sides:

2|3x - 5| = 16

Then, divide both sides by 2:

|3x - 5| = 8

Now, we consider two cases:

Case 1: 3x - 5 is positive
In this case, we can remove the absolute value signs:

3x - 5 = 8

Solving for x, we get:

3x = 13

x = 13/3

Case 2: 3x - 5 is negative
In this case, we can remove the absolute value signs and negate the expression:

-(3x - 5) = 8

Simplifying, we get:

-3x + 5 = 8

Solving for x, we get:

-3x = 3

x = -1

So, the correct answer is b.) x = -1 or x = 4 1/3

I hope this helps! Let me know if you have any further questions.

Of course, I'd be happy to help you with your algebra questions! Let's work through each question step by step:

1.) The sides of a triangle are in the ratio 3:4:5. If the triangle's perimeter is 90 cm, what is the length of each side?

To solve this problem, we need to find the length of each side. Let's set up the equation:
3x + 4x + 5x = 90
12x = 90
x = 7.5

Now, we can substitute the value of x back into the equation to find the lengths of the sides:
3(7.5) = 22.5 cm
4(7.5) = 30 cm
5(7.5) = 37.5 cm

Therefore, the correct answer is a.) 22.5 cm, 30 cm, 37.5 cm.

2.) Is the following equation sometimes, always, or never true?
6 + 8x - 9 = 11x + 14 - 3x

To determine the correctness of the equation, we can simplify both sides and see if they are equal:
6 + 8x - 9 = 11x + 14 - 3x
8x - 3 = 8x + 14

Here, we can observe that the variable 'x' is eliminated from the equation on both sides. This means that the equation is always true. So, the correct answer is b.) always true.

3.) Solve the inequality:
r is less than or equal to one fifth

To solve the inequality, we need to isolate 'r' on one side:
r ≤ 1/5

The inequality is already solved, and there is no need to manipulate it further. So, the correct answer is a.) r ≤ 1/5.

4.) Is the following inequality sometimes, always, or never true?
2(8x - 4) - 2x ≤ 14x + 12

To check the correctness of the inequality, let's simplify both sides:
16x - 8 - 2x ≤ 14x + 12
14x - 8 ≤ 14x + 12

Here, we can observe that the variable 'x' is eliminated from the inequality on both sides. This means that the inequality is always true. So, the correct answer is b.) always true.

5.) Solve the compound inequality:
6x - 4 < -34 or 3x + 10 > 10

We need to solve each inequality separately and then combine the solutions using the "or" operator (|):

6x - 4 < -34
6x < -30
x < -5

3x + 10 > 10
3x > 0
x > 0

Combining the solutions using the "or" operator (|), we get:
x < -5 or x > 0

So, the correct answer is c.) x < -5 or x > 0.

6.) Solve the inequality:
-2 ≤ 2x - 4 < 4

To solve this compound inequality, we need to isolate 'x' on the inequality:
-2 + 4 ≤ 2x - 4 + 4 < 4 + 4
2 ≤ 2x < 8

Now, let's divide the inequality by 2 to get the value of 'x':
1 ≤ x < 4

So, the correct answer is c.) 1 ≤ x < 4.

7.) Solve the absolute value equation:
2|3x - 5| - 8 = 8

To solve the equation, let's isolate the absolute value expression:
2|3x - 5| = 8 + 8
2|3x - 5| = 16

Now, divide both sides of the equation by 2 to solve for the absolute value:
|3x - 5| = 8

Next, we break the absolute value equation into two separate cases:

Case 1: 3x - 5 ≥ 0
When 3x - 5 ≥ 0, the absolute value equation becomes:
3x - 5 = 8
3x = 13
x = 13/3

Case 2: 3x - 5 < 0
When 3x - 5 < 0, the absolute value equation becomes:
-(3x - 5) = 8
-3x + 5 = 8
-3x = 3
x = -1

Therefore, the correct answer is d.) x = -1 or x = 13/3.

I hope the explanations provided above helped you understand how to solve these algebraic problems. If you have any further questions, please feel free to ask!