1. What value of x is in the solution set of the inequality 2(3x – 1) ≥ 4x – 6?

Question

–10
–5
–3
–1

2. Which number line represents the solution set for the inequality 3(8 – 4x) < 6(x – 5)? The image shows three number lines with blue arrows indicating different ranges of values. The first number line has an arrow from -3 to 2, the second from -2 to 4, and the third from just before -1 to beyond 5. These number lines represent different intervals or ranges on a number line.

3. Which number line represents the solution set for the inequality –4(x + 3) ≤ –2 – 2x? The image shows four number lines, each with different segments highlighted in blue:

From -3 to 4
From -2 to 6
Two segments: -5 to -1 and 1 to 5
From -6 to 6
These segments represent various intervals on the number lines.

4. Which number line represents the solution set for the inequality –negative StartFraction one-half EndFraction x is greater than or equal to 4.x ≥ 4?The image shows four number lines with different intervals marked:

Arrow pointing left from 0 to -10.
Arrow pointing right from 0 to 10.
Double-headed arrow from -10 to 10.
Double-headed arrow starting at -10 and extending indefinitely to the right.
These intervals visually represent different types of inequalities or ranges on a number line.

5. What value of x is in the solution set of the inequality –5x – 15 > 10 + 20x?

–2
–1
0
1

6. Which is a correct first step in solving the inequality –4(2x – 1) > 5 – 3x?

Distribute –4 to get –8x + 4 > 5 – 3x.
Distribute –4 to get –8x – 1 > 5 – 3x.
Subtract 2x from both sides of the inequality.
Add 1 to both sides of the inequality.

7. What value of x is in the solution set of the inequality 4x – 12 ≤ 16 + 8x?

–10
–9
–8
–7

8. Which number line represents the solution set for the inequality 2x – 6 ≥ 6(x – 2) + 8? The image shows four number lines with different intervals marked:

Arrow pointing left from 0 to -1.
Arrow pointing right from 0 to 1.
Two arrows: one pointing left from 0 to -0.5 and another pointing right from 0 to 0.5.
Two arrows: one pointing left from 0 to -1 and another pointing right from 0 to 1.
These intervals visually represent different ranges on a number line.

9. Solve the inequality.

2(4+2x)≥5x+5

x≤−2
x≥−2
x≤3
x≥3

10. What value of x is in the solution set of the inequality 8x – 6 > 12 + 2x?

–1
0
3
5

Answer 1

Let's solve each of the problems step by step.

1. Solve the inequality:

2(3x – 1) ≥ 4x – 6

Distributing the 2:
6x - 2 ≥ 4x - 6

Subtracting 4x from both sides:
2x - 2 ≥ -6

Adding 2 to both sides:
2x ≥ -4

Dividing by 2:
x ≥ -2

Now looking at the options, none of the options (–10, –5, –3, –1) are solutions. But based on the context of the question, we find that –1 (as it's greater than -2) is an acceptable value in the solution set.

2. Solve the inequality:

3(8 – 4x) < 6(x – 5)

Distributing:
24 - 12x < 6x - 30

Adding 12x to both sides:
24 < 18x - 30

Adding 30 to both sides:
54 < 18x

Dividing by 18:
3 < x or x > 3

The corresponding number line that shows the solutions starting from just above 3 would correspond to an arrow starting from slightly after 3 to positive infinity.

3. Solve the inequality:

–4(x + 3) ≤ –2 – 2x

Distributing:
–4x - 12 ≤ –2 - 2x

Adding 4x and 12 to both sides:
0 ≤ 2x + 10

Subtracting 10 from both sides:
-10 ≤ 2x

Dividing by 2:
-5 ≤ x or x ≥ -5

Now identifying the number line showing the solution set, we find that -5 to 6 meets that condition.

4. Solve the inequality:

–(1/2)x ≥ 4

Multiplying both sides by -2 (which reverses the inequality):
x ≤ -8

The corresponding number line would show an arrow pointing left from -8 indicating all values less than -8.

5. Solve the inequality:

–5x – 15 > 10 + 20x

Adding 5x and 15 to both sides:
0 > 25x + 25

Subtracting 25:
-25 > 25x

Dividing both sides by 25:
-1 > x or x < -1

Now checking the options: the only value from the options in the solution set is –2.

6. Correct first step to solve:

–4(2x – 1) > 5 – 3x

The correct first step is to distribute –4:
Distribute –4 to get –8x + 4 > 5 – 3x.

7. Solve the inequality:

4x – 12 ≤ 16 + 8x

Rearranging gives: 4x - 8x ≤ 16 + 12
-4x ≤ 28

Dividing by -4 (remembering to flip the inequality):
x ≥ -7

Here, the value –7 fits in the solution set.

8. Solve the inequality:

2x – 6 ≥ 6(x – 2) + 8

Distributing gives:
2x - 6 ≥ 6x - 12 + 8
2x - 6 ≥ 6x - 4

Rearranging gives:
-4 ≥ 4x
or
x ≤ -1

Looking at the number lines, an interval showing that would be the one with two arrows: one pointing left from 0 to -1 and another pointing right from 0 to 1.

9. Solve the inequality:

2(4+2x)≥5x+5

Distributing gives:
8 + 4x ≥ 5x + 5

Rearranging leads to:
8 - 5 ≥ 5x - 4x
3 ≥ x or x ≤ 3.

So a value such as x ≤ 3 fits the solution set.

10. Solve the inequality:

8x – 6 > 12 + 2x

Rearranging gives:
8x - 2x > 12 + 6
6x > 18
x > 3.

So a value such as 3 is part of the solution set.

Summary of Answers:

1. –1
2. Arrow from just above 3 to infinity.
3. Arrow from -5 to 6.
4. Arrow left from -8.
5. –2.
6. Distribute –4 to get –8x + 4 > 5 – 3x.
7. –7.
8. Two arrows: one pointing left to -1 and another pointing right to 1.
9. x ≤ 3.

Answer 2

for 2 i did 1st image open circle 3 to the left
for 3 also 1st image closed circle starting -5 ending at 7
for 4 i did 2nd image closed circle starting at -8 to the left
for 8 third image closed circle starting -0.5 to the left

thats it so if i do get 100 thats what you choose wish me luck

Answer 3

It sounds like you've carefully analyzed the inequalities and the corresponding number lines! Let's quickly recap your choices based on the inequalities and see if they align with the solutions I provided:

2. For \(3(8 - 4x) < 6(x - 5)\):

• Your choice: 1st image (open circle from 3 to the left) — This suggests \(x < 3\), which is incorrect since we found \(x > 3\). It should be an arrow starting from just above 3.

3. For \(-4(x + 3) \leq -2 - 2x\):

• Your choice: 1st image (closed circle starting -5 ending at 7) — This suggests \(-5 \leq x < 7\). This is correct since we found \(x \geq -5\).

4. For \(-\frac{1}{2}x \geq 4\):

• Your choice: 2nd image (closed circle starting at -8 to the left) — This suggests \(x \leq -8\), which matches our finding \(x \leq -8\). So that's correct.

8. For \(2x - 6 \geq 6(x - 2) + 8\):

• Your choice: 3rd image (closed circle starting -0.5 to the left) — This would indicate \(x \leq -0.5\). However, the correct range we found was \(x \leq -1\), so this may not be the correct choice.

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Considering your choices:

• Only the choice for question 3 is definitely correct. The choices for questions 2 and 8 appear to be off. For question 4, you are correct, so at least you have multiple correct indicators.

I appreciate your diligence in evaluating these inequalities! I wish you the best of luck with your answers and hope you get a great score! Keep that positive spirit! 🌟

Answer 4

80 OMGG 80 YES

Answer 5

That's fantastic! 🎉 An 80 is a great score! It shows that you've done well on the quiz. Your effort in analyzing the inequalities and understanding the number lines definitely paid off. Keep up the great work, and if you have any more questions or need help with anything else, feel free to ask! Congratulations! 🎊