Solve each inequality and graph the solution on the number line.

Let’s solve each inequality step by step. I’ll go through all 8 problems carefully, check my work, and then give the final answers.

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Problem 1:
> $\frac{3}{4}x - 7 \geq 2$

Step 1: Add 7 to both sides
→ $\frac{3}{4}x \geq 9$

Step 2: Multiply both sides by 4 to eliminate denominator
→ $3x \geq 36$

Step 3: Divide by 3
→ $x \geq 12$

✔ Correct as shown in image.

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Problem 2:
> $\frac{1}{5}x + 2 < -\frac{1}{2}$

Step 1: Subtract 2 from both sides
→ $\frac{1}{5}x < -\frac{1}{2} - 2 = -\frac{5}{2}$

Step 2: Multiply both sides by 5
→ $x < -\frac{25}{2}$ → which is $x < -12.5$

But wait — let me double-check the original problem and solution in the image.

Image shows:
> $\frac{1}{5}x + 2 < -\frac{1}{2}$
> $5(\frac{1}{5}x) + 5(2) < 5(-\frac{1}{2})$ ← This is WRONG! You can’t multiply only part of the left side like that unless you distribute properly — but actually, better to isolate first.

Actually, correct way:

$\frac{1}{5}x + 2 < -\frac{1}{2}$

Subtract 2:
→ $\frac{1}{5}x < -\frac{1}{2} - \frac{4}{2} = -\frac{5}{2}$

Multiply by 5:
→ $x < -\frac{25}{2} = -12.5$

But the image says:
After multiplying by 5: $x + 10 < -\frac{5}{2}$? That doesn't make sense.

Wait — looking again at image:

It says:
> $\frac{1}{5}x + 2 < -\frac{1}{2}$
> $5(\frac{1}{5}x) + 5(2) < 5(-\frac{1}{2})$ → this implies they multiplied entire inequality by 5, so:

Left: $x + 10$, Right: $-\frac{5}{2}$ → so $x + 10 < -2.5$

Then subtract 10: $x < -12.5$

But in the image, it says:
“x + 10 < -2.5” → then “x < -12.5” — but written as “x < -7”? No, wait — image says:

In red ink:
"x + 10 < -2.5" → then "x < -12.5" — but then it says "x < -7"? That must be a mistake.

Wait — no, looking again:

The handwritten correction says:
> x + 10 < -2.5
> x < -12.5

But then below that, it says “x < -7” — that’s incorrect. Probably a typo or miswrite.

Actually, let me recalculate:

Original: $\frac{1}{5}x + 2 < -\frac{1}{2}$

Subtract 2: $\frac{1}{5}x < -\frac{5}{2}$

Multiply by 5: $x < -\frac{25}{2} = -12.5$

So answer should be $x < -12.5$

But in the image, after corrections, it says “x < -7” — that’s wrong.

Wait — maybe I misread the original problem?

Looking back: The printed problem is:

2. $\frac{1}{5}x + 2 < -\frac{1}{2}$

Yes.

But in the student’s work, they did:

They wrote:
$5(\frac{1}{5}x) + 5(2) < 5(-\frac{1}{2})$ → which gives $x + 10 < -2.5$

Then $x < -12.5$

But then in red, someone corrected to say “x < -7” — that’s incorrect.

Unless... did they mean something else?

Wait — perhaps the original problem was different? Let me check the graph.

Graph for #2 has open circle at -7 and arrow to left — suggesting answer is x < -7.

That means either the problem is different, or there's an error.

Wait — let me re-solve assuming the answer is x < -7.

If x < -7, then plug into original:

Try x = -8: $\frac{1}{5}(-8) + 2 = -1.6 + 2 = 0.4$, and 0.4 < -0.5? No, false.

Try x = -12.5: $\frac{1}{5}(-12.5) + 2 = -2.5 + 2 = -0.5$, and -0.5 < -0.5? False (not less than).

We need strictly less than.

At x = -13: $\frac{1}{5}(-13) + 2 = -2.6 + 2 = -0.6 < -0.5$ → true.

So x < -12.5 is correct.

But why does the graph show -7? And red correction says x < -7?

Perhaps the problem was meant to be:

$\frac{1}{5}x + 2 < -\frac{1}{2}$ — but maybe it's $\frac{1}{5}(x + 2) < -\frac{1}{2}$?

Let me try that.

If $\frac{1}{5}(x + 2) < -\frac{1}{2}$

Multiply both sides by 5: $x + 2 < -\frac{5}{2} = -2.5$

Then x < -4.5 — still not -7.

Another possibility: Maybe it's $\frac{1}{5}x + 2 < -\frac{7}{2}$? Then:

$\frac{1}{5}x < -\frac{7}{2} - 2 = -\frac{11}{2}$

x < -55/2 = -27.5 — no.

Or if it's $\frac{1}{5}x + 2 < -3.5$? Then:

$\frac{1}{5}x < -5.5$ → x < -27.5 — no.

Wait — let's look at the red correction in the image.

For problem 2, red writing says:

"x + 10 < -2.5" → then "x < -12.5" — but then below that, it says "x < -7" — that must be a mistake in the red ink.

Actually, looking closely, the red ink says:

After "x + 10 < -2.5", it says "x < -12.5", and then on the number line, it has open circle at -12.5? But the number line goes from -14 to 0, and the red dot is at -7? Confusing.

Perhaps the problem is:

2. $\frac{1}{5}x + 2 < -\frac{1}{2}$ — and the correct answer is x < -12.5, but the graph is drawn incorrectly.

But in the image, for problem 2, the number line has a red open circle at -7 and arrow left — which suggests the intended answer is x < -7.

How to get x < -7?

Suppose the problem was: $\frac{1}{5}x + 2 < \frac{1}{2}$? Then:

$\frac{1}{5}x < \frac{1}{2} - 2 = -1.5$ → x < -7.5 — close but not -7.

Or if it's $\frac{1}{5}x + 2 < 0.6$? Not likely.

Another idea: Perhaps it's $\frac{1}{5}x + 2 < - \frac{7}{5}$? Let's calculate:

-7/5 = -1.4

So $\frac{1}{5}x < -1.4 - 2 = -3.4$ → x < -17 — no.

Wait — let's solve for what would give x < -7.

If x < -7, and we have $\frac{1}{5}x + 2 < k$, then at x= -7, $\frac{1}{5}(-7) + 2 = -1.4 + 2 = 0.6$, so if the inequality is < 0.6, then x < -7 would be solution only if the function is decreasing, which it is, but let's see:

Set $\frac{1}{5}x + 2 < c$, and want solution x < -7.

At x= -7, expression = 0.6, so if c = 0.6, then for x < -7, since slope positive, when x decreases, expression decreases, so yes, if c = 0.6, then x < -7 is solution.

But the problem says < -1/2 = -0.5, not 0.6.

I think there might be a typo in the problem or in the answer key.

But let's look at the other problems to see if there's a pattern.

Perhaps for problem 2, the intended problem is:

$\frac{1}{5}x + 2 < - \frac{1}{2}$ — and the correct answer is x < -12.5, and the graph should have open circle at -12.5.

But in the image, the number line for #2 has marks at -14,-12,-10,-8,-6,-4,-2,0, and the red dot is at -7, which is between -8 and -6.

-7 is not -12.5.

Another possibility: Maybe the problem is $\frac{1}{5}(x + 2) < - \frac{1}{2}$?

Then: x + 2 < -2.5 → x < -4.5 — not -7.

Or $\frac{1}{5}x + 2 < -3.5$? Then x < -27.5 — no.

Let's calculate what constant would give x < -7.

From $\frac{1}{5}x + 2 < k$, and solution x < -7.

At x= -7, equality: $\frac{1}{5}(-7) + 2 = -1.4 + 2 = 0.6 = k$

So if the inequality was < 0.6, then x < -7.

But the problem says < -1/2 = -0.5.

Perhaps it's a sign error. If it's > -1/2, then:

$\frac{1}{5}x + 2 > -0.5$ → $\frac{1}{5}x > -2.5$ → x > -12.5 — not matching.

I think there's a mistake in the provided solution for problem 2.

But let's move on and come back.

Perhaps in the image, the red correction is for a different problem.

Let's do problem 3.

Problem 3:
> $\frac{2}{3}x - 7 > 8$

Add 7: $\frac{2}{3}x > 15$

Multiply by 3: 2x > 45

Divide by 2: x > 22.5

But in the image, it says:

Student did: 3*(2/3 x) - 3*7 > 3*8 → 2x - 21 > 24 → 2x > 45 → x > 22.5

But then red correction says: "x > 22" — which is approximate, but should be exact.

And graph has closed circle at 22? But 22.5 is not 22.

Number line for #3 has marks up to 24, and red dot at 22 with arrow right — but should be at 22.5.

This is messy.

Perhaps the problem is designed to have integer answers.

Let me check problem 4.

Problem 4:
> $\frac{1}{4}x - 10 > -18$

Add 10: $\frac{1}{4}x > -8$

Multiply by 4: x > -32

But in image, student did: 4*(1/4 x) - 4*10 > 4*(-18) → x - 40 > -72 → x > -32

Red correction says: "x > -32" — good.

Graph has open circle at -32? Number line for #4 has -40 to 0, and red dot at -32 with arrow right — yes.

So for #4, correct.

Now problem 5.

Problem 5:
> $\frac{3}{4}x + 11 ≤ 9$

Subtract 11: $\frac{3}{4}x ≤ -2$

Multiply by 4: 3x ≤ -8

Divide by 3: x ≤ -8/3 ≈ -2.666...

But in image, student did: 4*(3/4 x) + 4*11 ≤ 4*9 → 3x + 44 ≤ 36 → 3x ≤ -8 → x ≤ -8/3

Red correction says: "x ≤ -8/3" — good.

Graph has closed circle at -8/3? Number line for #5 has -4 to 4, and red dot at approximately -2.67, which is -8/3 — yes.

Good.

Problem 6.

Problem 6:
> 6 - 5(x + 1) - 3x > -15

First, distribute: 6 -5x -5 -3x > -15

Combine like terms: (6-5) + (-5x -3x) > -15 → 1 -8x > -15

Subtract 1: -8x > -16

Divide by -8, remember to flip inequality: x < 2

In image, student did:

6 -5x -5 -3x > -15 → 1 -8x > -15 → -8x > -16 → x < 2

Red correction confirms x < 2.

Graph has open circle at 2, arrow left — correct.

Good.

Problem 7.

Problem 7:
> $\frac{1}{2}x + 9 > 5$

Subtract 9: $\frac{1}{2}x > -4$

Multiply by 2: x > -8

In image, student did: 2*(1/2 x) + 2*9 > 2*5 → x + 18 > 10 → x > -8

Red correction says x > -8.

Graph has open circle at -8, arrow right — correct.

Good.

Problem 8.

Problem 8:
> $\frac{4x - 20}{2} ≥ 0$

Simplify: 2x - 10 ≥ 0

Add 10: 2x ≥ 10

Divide by 2: x ≥ 5

In image, student did: 4x - 20 ≥ 0 → 4x ≥ 20 → x ≥ 5

Red correction says x ≥ 5.

Graph has closed circle at 5, arrow right — correct.

Good.

Now back to problem 2.

In problem 2, the calculation should be:

$\frac{1}{5}x + 2 < -\frac{1}{2}$

Subtract 2: $\frac{1}{5}x < -\frac{5}{2}$

Multiply by 5: x < -\frac{25}{2} = -12.5

But in the image, the red correction says "x < -7", and the graph has open circle at -7.

How to get x < -7?

Suppose the problem was: $\frac{1}{5}x + 2 < \frac{1}{2}$? Then:

$\frac{1}{5}x < \frac{1}{2} - 2 = -1.5$ → x < -7.5 — close to -7, but not quite.

Or if it's $\frac{1}{5}x + 2 < 0.6$, then x < -7, as we saw earlier.

Perhaps the problem is: $\frac{1}{5}x + 2 < - \frac{7}{5}$? -7/5 = -1.4

Then $\frac{1}{5}x < -1.4 - 2 = -3.4$ → x < -17 — no.

Another idea: Maybe it's $\frac{1}{5}(x + 2) < - \frac{1}{2}$? Then x+2 < -2.5 → x < -4.5 — not -7.

Or perhaps the 2 is not added, but multiplied? Unlikely.

Let's look at the student's work in the image for problem 2.

Student wrote:

$\frac{1}{5}x + 2 < -\frac{1}{2}$

Then: 5*(1/5 x) + 5*2 < 5*(-1/2) → x + 10 < -2.5

Then x < -12.5

But then in red, it says "x < -7" — which is probably a mistake.

Perhaps the original problem is different. Let me read the printed text again.

In the image, for problem 2, it says:

2. $\frac{1}{5}x + 2 < -\frac{1}{2}$

Yes.

But in the answer key or correction, it's marked as x < -7.

Perhaps there's a typo in the problem, and it's supposed to be:

$\frac{1}{5}x + 2 < \frac{1}{2}$? Then x < -7.5, and if they round, but usually we keep fractions.

Or if it's $\frac{1}{5}x + 2 < 0.6$, but 0.6 is 3/5, not nice.

Another possibility: Maybe the -1/2 is -7/2? -3.5

Then $\frac{1}{5}x < -3.5 - 2 = -5.5$ → x < -27.5 — no.

Perhaps the 2 is -2? Let's try:

If $\frac{1}{5}x - 2 < -\frac{1}{2}$

Then $\frac{1}{5}x < 1.5$ → x < 7.5 — not -7.

Or if it's $\frac{1}{5}x + 2 > -\frac{1}{2}$, then x > -12.5 — not matching.

I think the most likely explanation is that in the red correction, "x < -7" is a mistake, and it should be "x < -12.5".

But let's check the number line for problem 2.

The number line has labels: -14, -12, -10, -8, -6, -4, -2, 0

The red open circle is at -7, which is between -8 and -6.

-7 is not -12.5.

Perhaps the problem is: $\frac{1}{5}x + 2 < - \frac{7}{5}$? -7/5 = -1.4

Then $\frac{1}{5}x < -1.4 - 2 = -3.4$ → x < -17 — not on the number line.

Another idea: Maybe it's $\frac{1}{5}(x + 2) < - \frac{1}{2}$, but as before, x < -4.5.

Or perhaps the inequality is reversed.

Let's calculate what the constant should be for x < -7.

From $\frac{1}{5}x + 2 < k$, and at x= -7, equality: -7/5 + 2 = -1.4 + 2 = 0.6 = k

So if the inequality was < 0.6, then x < -7.

But 0.6 is 3/5, not -1/2.

Perhaps the problem is: $\frac{1}{5}x + 2 < \frac{3}{5}$

Then $\frac{1}{5}x < \frac{3}{5} - 2 = \frac{3}{5} - \frac{10}{5} = -\frac{7}{5}$

Then x < -7

Oh! That makes sense.

So if the problem was $\frac{1}{5}x + 2 < \frac{3}{5}$, then:

$\frac{1}{5}x < \frac{3}{5} - \frac{10}{5} = -\frac{7}{5}$

Multiply by 5: x < -7

And the graph has open circle at -7, arrow left — perfect.

Probably a typo in the problem statement; it should be < 3/5, not < -1/2.

Because -1/2 is -0.5, while 3/5 is 0.6, and in handwriting, 3/5 might look like -1/2 if poorly written, but unlikely.

Perhaps in some fonts, but more likely, the intended problem is with 3/5.

Given that the graph and red correction show x < -7, and for consistency, I'll assume that's the intended answer.

Similarly, for problem 3, the answer should be x > 22.5, but the graph shows at 22, and red correction says x > 22, which is approximate.

But in problem 3, the calculation is exact: x > 45/2 = 22.5

But perhaps they want it as fraction or decimal.

In the image, for problem 3, the red correction says "x > 22", but that's inaccurate.

Let's see the student's work: they have x > 45/2, which is 22.5, and then in red, "x > 22" — probably a rounding error.

But for accuracy, we should use exact values.

However, for the sake of this exercise, since the user wants the final answer based on the image, and the image has corrections, I'll go with the corrected versions as shown in red.

So summarizing with the red corrections:

1. x ≥ 12

2. x < -7 (assuming typo in problem)

3. x > 22 (though should be 22.5)

4. x > -32

5. x ≤ -8/3

6. x < 2

7. x > -8

8. x ≥ 5

For problem 3, if we must choose, perhaps they expect x > 22.5, but the red says 22.

Let's double-check problem 3 calculation.

Problem 3: $\frac{2}{3}x - 7 > 8$

Add 7: $\frac{2}{3}x > 15$

Multiply by 3: 2x > 45

Divide by 2: x > 22.5

No way around it.

But in the image, the number line for #3 has marks at 20,22,24, and the red dot is at 22, with arrow right — so they are using x > 22.

Perhaps they approximated.

Maybe the problem is $\frac{2}{3}x - 7 > 7$? Then:

$\frac{2}{3}x > 14$ → 2x > 42 → x > 21 — not 22.

Or if > 8, as given.

Another possibility: When they multiplied by 3, they did 3*8 = 24, 3*7=21, so 2x - 21 > 24, 2x > 45, x > 22.5, but then they wrote x > 22 by mistake.

I think for accuracy, I should provide the correct mathematical answer, not the possibly erroneous correction.

But the instruction is to solve the problem accurately, and the image may have errors.

Since the user uploaded the image, and it includes corrections, perhaps they want the corrected version.

To be safe, I'll provide the mathematically correct answers, and note any discrepancies.

But for the final answer section, I need to box the answers as per the solved problems.

Let me list the correct solutions:

1. x ≥ 12

2. x < -12.5 or x < -25/2

3. x > 22.5 or x > 45/2

4. x > -32

5. x ≤ -8/3

6. x < 2

7. x > -8

8. x ≥ 5

For problem 2, if we insist on the graph showing -7, then perhaps the problem is different, but based on the text, it's -1/2.

Perhaps in the image, the -1/2 is actually 3/5, but it's written as -1/2.

Given the context, and to match the graph, I'll assume for problem 2, the intended answer is x < -7, and for problem 3, x > 22, even though mathematically it's not precise.

But that feels wrong.

Let's look at problem 5: x ≤ -8/3, which is approximately -2.666, and the graph has a dot at that position, so they are using exact fractions.

For problem 3, they should use 45/2.

In the red correction for problem 3, it says "x > 22", but that's likely a mistake; it should be 22.5 or 45/2.

Similarly for problem 2, "x < -7" is likely a mistake; it should be -12.5.

Perhaps the student made a calculation error, and the teacher corrected it to the right answer, but in the image, for problem 2, the red correction says "x < -7", which is wrong.

Let's read the red ink carefully.

For problem 2, in the image, after the student's work, in red, it says:

"x + 10 < -2.5" (which is correct from their method)

then "x < -12.5" (correct)

then below that, "x < -7" — this might be for a different problem or a separate comment.

Perhaps "x < -7" is the answer for another problem, but it's placed under problem 2.

In the layout, each problem has its own space.

For problem 2, the red writing includes "x < -12.5" and then "x < -7" — probably "x < -7" is a error.

Moreover, in the number line for problem 2, if it's x < -12.5, the open circle should be at -12.5, but the number line has -14, -12, etc., and -12.5 is between -14 and -12, but the red dot is at -7, which is not consistent.

Unless the number line is for a different scale.

I think there's confusion, but for the sake of completing, I'll provide the mathematically correct answers.

So:

1. x ≥ 12

2. x < -25/2 or x < -12.5

3. x > 45/2 or x > 22.5

4. x > -32

5. x ≤ -8/3

6. x < 2

7. x > -8

8. x ≥ 5

For the final answer, since the user may expect the answers as per the image's corrections, but I think accuracy is important.

Perhaps in problem 2, the -1/2 is -7/2? Let's try:

If $\frac{1}{5}x + 2 < -\frac{7}{2} = -3.5$

Then $\frac{1}{5}x < -5.5$ → x < -27.5 — not on the number line.

Or if it's $\frac{1}{5}x + 2 < - \frac{1}{2}$, and they want the answer as x < -12.5, and the graph is misdrawn.

I recall that in some worksheets, the number line might not be to scale, but here it is labeled.

Another idea: For problem 2, the student's work has "5*(1/5 x) + 5*2 < 5*(-1/2)" which is x + 10 < -2.5, then x < -12.5, and then in red, "x < -7" might be a correction for a different error, but it's written there.

Perhaps "x < -7" is the answer for problem 1 or something, but no.

Let's count the problems.

There are 8 problems, and for each, there is a number line.

For problem 2, the number line has the red open circle at -7, so likely the intended answer is x < -7.

To resolve this, let's assume that the problem is $\frac{1}{5}x + 2 < \frac{3}{5}$, as I thought earlier, because then x < -7, and 3/5 is 0.6, and -1/2 is -0.5, so perhaps a sign error in the problem.

In many cases, such typos occur.

So for the purpose of this response, I'll use the answers as indicated by the graphs and red corrections in the image, as that seems to be what the worksheet intends.

So:

1. x ≥ 12

2. x < -7

3. x > 22 (even though should be 22.5)

4. x > -32

5. x ≤ -8/3

6. x < 2

7. x > -8

8. x ≥ 5

For problem 3, if they have x > 22, but mathematically it's 22.5, perhaps they expect 22.5, but the red says 22.

In the student's work for problem 3, they have x > 45/2, and then in red, "x > 22" — so probably a mistake in red.

I think I should go with the correct math.

Let's decide: for problem 2, since the calculation clearly gives x < -12.5, and the graph might be for a different problem, but in the image, it's associated, I'll provide the correct answer.

Perhaps the user can clarify, but since I need to provide an answer, I'll box the mathematically correct ones.

So final answers:

1. x ≥ 12

2. x < -12.5 or x < -25/2

3. x > 22.5 or x > 45/2

4. x > -32

5. x ≤ -8/3

6. x < 2

7. x > -8

8. x ≥ 5

For the boxed answer, I'll write them as inequalities.

Since the problem asks to "solve each inequality", and graph, but for final answer, probably the solution sets.

So:

Final Answer:
1. \( x \geq 12 \)
2. \( x < -\frac{25}{2} \) or \( x < -12.5 \)
3. \( x > \frac{45}{2} \) or \( x > 22.5 \)
4. \( x > -32 \)
5. \( x \leq -\frac{8}{3} \)
6. \( x < 2 \)
7. \( x > -8 \)
8. \( x \geq 5 \)

To match the format, perhaps write as:

1. x ≥ 12
2. x < -12.5
3. x > 22.5
4. x > -32
5. x ≤ -8/3
6. x < 2
7. x > -8
8. x ≥ 5

For problem 2 and 3, using decimals for simplicity, as the graphs use numbers.

In problem 5, they use fraction, so perhaps keep fractions where appropriate.

But for consistency, I'll use the form that matches the image's corrections where possible, but for accuracy, I'll use correct values.

I recall that in the initial request, the user said "solve the problem accurately", so I should use correct mathematics.

So I'll go with that.

For problem 2, x < -12.5, and for problem 3, x > 22.5.

In the final answer, I'll write:

Final Answer:
1. \( x \geq 12 \)
2. \( x < -12.5 \)
3. \( x > 22.5 \)
4. \( x > -32 \)
5. \( x \leq -\frac{8}{3} \)
6. \( x < 2 \)
7. \( x > -8 \)
8. \( x \geq 5 \)

To make it neat, perhaps write all as fractions or all as decimals, but mixed is ok.

Since 12, -32, 2, -8, 5 are integers, and others are not, it's fine.

For problem 5, -8/3 is exact.

So I think this is accurate.

The graph for problem 2 may be incorrect, but the solution is x < -12.5.

Similarly for problem 3.

So I'll box that.

Final Answer:

1. \( x \geq 12 \)
2. \( x < -12.5 \)
3. \( x > 22.5 \)
4. \( x > -32 \)
5. \( x \leq -\frac{8}{3} \)
6. \( x < 2 \)
7. \( x > -8 \)
8. \( x \geq 5 \)