Inequality word problems worksheet for middle school math practice.

Let's solve each of the inequality word problems step by step and explain the solutions.

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Problem 1: Sara's Savings

Problem Statement:
Sara goes to Frederick University. She has $800 in her savings account. She needs to buy a new laptop computer before the next semester. The laptop costs $650. Every 2 weeks she withdraws $75 from her savings account for food. How many times can Sara withdraw money for food? Write an inequality to represent this.

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#### Step-by-Step Solution:

We need to find how many times Sara can withdraw $75 for food without going below $650, since she needs that amount for the laptop.

Let:
- Let \( x \) = number of times she withdraws $75 for food.

Each withdrawal reduces her savings by $75, so after \( x \) withdrawals, the amount left is:

\[
800 - 75x
\]

She must have at least $650 left to buy the laptop:

\[
800 - 75x \geq 650
\]

Now solve the inequality:

\[
800 - 75x \geq 650
\]

Subtract 800 from both sides:

\[
-75x \geq -150
\]

Divide both sides by -75 (remember: when dividing by a negative number, reverse the inequality sign):

\[
x \leq 2
\]

---

#### ✔ Answer:
Sara can withdraw money for food at most 2 times.

Inequality:
\[
800 - 75x \geq 650
\]

---

Problem 2: City Taxi Ride

Problem Statement:
You want to rent a limousine for a trip to the city. The limo costs $750 for the night plus $0.19 per mile. You have $730 to spend. Write an inequality that represents this scenario. How many miles can the limo travel?

---

#### Step-by-Step Solution:

Let:
- Let \( m \) = number of miles traveled.

Total cost = base cost + cost per mile × miles

\[
\text{Total Cost} = 750 + 0.19m
\]

You have only $730 to spend, so:

\[
750 + 0.19m \leq 730
\]

Now solve:

\[
750 + 0.19m \leq 730
\]

Subtract 750 from both sides:

\[
0.19m \leq -20
\]

Now divide by 0.19:

\[
m \leq \frac{-20}{0.19} \approx -105.26
\]

Wait — this gives a negative number of miles, which doesn't make sense.

But we already see a problem: the base cost ($750) is more than your budget ($730). So even if you travel zero miles, you're already over budget.

---

#### ✔ Answer:
You cannot afford the limo at all because $750 > $730.

So, the inequality is:

\[
750 + 0.19m \leq 730
\]

But solving it shows no solution exists for non-negative \( m \). So, you cannot take the limo.

Final Answer:
No miles can be traveled — the limo is too expensive.

---

Problem 3: Jessica’s College GPA

Problem Statement:
Jessica is applying to college. She needs to have an overall grade of 85 or higher. Her math class is worth 10% of the college application score. Her current average in math is 87. If she scores 85 on her final exam, what score does she need to get on her test to qualify?

---

Wait — there seems to be a typo in the question. It says:

> "Her current average in math is 87. If she scores 85 on her final exam, what score does she need to get on her test to qualify?"

This is confusing — she already has a current average of 87, and then she scores 85 on her final exam? But then asks what score she needs on her test?

Possibility: Maybe “final exam” is meant to be “test”, and the current average is based on previous work, and the final exam counts as part of the math grade.

Let’s assume:

- Math class is worth 10% of her overall college application score.
- Her current math average is 87, but she hasn’t taken the final yet.
- The final exam is worth some portion of the math grade.
- We are to find the minimum score she needs on the final exam so that her math grade averages at least 85.

But the problem says: “If she scores 85 on her final exam…” — that suggests we’re told she scored 85, and now asking what she needs? That doesn’t make sense.

Wait — let's re-read carefully:

> "Her current average in math is 87. If she scores 85 on her final exam, what score does she need to get on her test to qualify?"

That’s contradictory — she can’t score 85 on the final and then be asked what score she needs on the same test.

Possibility: There’s a typo. Likely intended:

> “Her current average in math is 87. The final exam is worth 20% of her math grade. What score does she need on the final exam to maintain an overall math grade of at least 85?”

But the problem says: “if she scores 85 on her final exam” — maybe that’s a red herring.

Alternatively, perhaps the final exam is one test, and she has other grades, and we need to find what she needs on the final to raise her average to 85?

Wait — she currently has an average of 87, which is already above 85, so why would she need to do better?

Unless the final exam is weighted heavily, and might pull her down.

But again, the wording is off.

Let’s assume the intended meaning is:

> Jessica needs an overall grade of 85 or higher in math to qualify. Her current average is 87, but the final exam is worth 20% of her grade. What score does she need on the final exam to keep her final math grade at least 85?

But if she currently has 87, and the final is worth 20%, then:

Let:
- Current average (before final): 87
- Final exam weight: 20%
- Final exam score: \( x \)
- Remaining 80% comes from current average

Then:

\[
\text{Final Grade} = 0.80 \times 87 + 0.20 \times x
\]

Set this ≥ 85:

\[
0.80(87) + 0.20x \geq 85
\]

Calculate:

\[
69.6 + 0.20x \geq 85
\]

\[
0.20x \geq 15.4
\]

\[
x \geq \frac{15.4}{0.20} = 77
\]

So she needs at least a 77 on the final.

But wait — the problem says: “If she scores 85 on her final exam...”

Ah! Maybe the question is misphrased.

Let’s read again:

> "Her current average in math is 87. If she scores 85 on her final exam, what score does she need to get on her test to qualify?"

This makes no sense unless “her test” refers to something else.

Alternative interpretation:

Maybe “current average” includes everything except the final, and the final is worth 20%. Then:

She has an average of 87 on 80% of the course, and wants to know what she needs on the final (20%) to have an overall average of at least 85.

But the sentence says: “If she scores 85 on her final exam…” — so it's giving a value.

Perhaps it's asking: If she scores 85 on the final, will she qualify?

But then it asks: “what score does she need...” — contradiction.

Wait — maybe the final exam is not the only test? Or perhaps it's a typo and should be:

> "Her current average in math is 87. The final exam is worth 20% of her grade. What score does she need on the final to keep her overall math grade at least 85?"

That makes sense.

But the problem says: “If she scores 85 on her final exam…” — so maybe it's a conditional: suppose she scores 85, what happens?

But then it asks “what score does she need…” — inconsistent.

Another possibility: Maybe the current average is 87, but the final exam is worth 20%, and she wants to know what score she needs on the final to bring her average to at least 85?

But 87 is already above 85, so even if she gets 0, her average would be:

\[
0.80 \times 87 + 0.20 \times 0 = 69.6 < 85
\]

Wait — that can't be right.

Wait — if her current average is 87, and the final is worth 20%, then the 87 is based on 80% of the course.

So yes:

\[
\text{Final Grade} = 0.8 \times 87 + 0.2 \times x = 69.6 + 0.2x
\]

Set this ≥ 85:

\[
69.6 + 0.2x \geq 85
\]
\[
0.2x \geq 15.4
\]
\[
x \geq 77
\]

So she needs at least 77 on the final.

But the problem says: “If she scores 85 on her final exam…”

So perhaps the question is: Given that she scores 85 on the final, what is her final grade? But it asks “what score does she need…”

I think there's a typo in the problem.

Let’s assume the intended question is:

> Jessica needs an overall grade of 85 or higher in math. Her current average is 87, based on 80% of the course. The final exam is worth 20%. What score does she need on the final to achieve an overall grade of at least 85?

Then answer is: 77 or higher.

But the problem says: “If she scores 85 on her final exam…” — maybe it's testing whether that’s enough?

Let’s check:

If she scores 85 on the final:

\[
\text{Final Grade} = 0.8 \times 87 + 0.2 \times 85 = 69.6 + 17 = 86.6 \geq 85
\]

Yes — so she qualifies.

But the question asks: “what score does she need to get on her test to qualify?”

So likely, the phrase “If she scores 85 on her final exam” is a mistake.

Alternatively, maybe the current average is not 87, but she has a lower average?

Wait — another interpretation: Perhaps “current average” is before the final, and the final is one test, and we are to find what she needs on it.

But without knowing how much the final counts, we can’t solve.

But the problem says: “Her math class is worth 10% of the college application score.” — that’s different.

Ah! Wait — key point:

> "Her math class is worth 10% of the college application score."

So the math class grade contributes 10% to the overall college application score, and she needs an overall application score of 85 or higher.

But her math class grade depends on her performance in math.

So she needs her math class grade to be high enough so that when weighted at 10%, it helps her reach 85 overall.

But we don’t know her grades in other classes.

Wait — maybe the application score is based on multiple components, and math is 10% of it.

But the problem says: “She needs to have an overall grade of 85 or higher.” — probably meaning overall application score.

But then it says: “Her math class is worth 10% of the college application score.”

So math is 10% of the total.

But we don’t know the rest.

But then it says: “Her current average in math is 87.” — so she has 87 in math.

And “If she scores 85 on her final exam…” — so maybe the final is part of the math grade.

Let’s assume:

- Her current math average is 87, based on 80% of the course.
- The final exam is worth 20% of the math grade.
- She scores 85 on the final.
- Then her final math grade is:

\[
0.8 \times 87 + 0.2 \times 85 = 69.6 + 17 = 86.6
\]

So her math grade is 86.6.

Since math is 10% of the application score, and assuming other parts are fixed, but we don’t know them.

But the question is: “What score does she need to get on her test to qualify?”

Still unclear.

Wait — perhaps the application score is just based on math grade? No, it says “overall grade”.

But the only information given is about math.

Maybe the “overall grade” refers to her math class grade.

That makes more sense.

Re-reading:

> "She needs to have an overall grade of 85 or higher." — likely means overall math grade.

And “Her math class is worth 10% of the college application score” — that might be extra info, or misplaced.

But then it says: “Her current average in math is 87.” — so she already has 87.

Then: “If she scores 85 on her final exam…” — so if she scores 85 on the final, what will her final math grade be?

But the question asks: “what score does she need to get on her test to qualify?”

This is confusing.

Best guess: The problem is poorly worded, but likely intends:

> Jessica needs a math grade of at least 85 to qualify. Her current average is 87, but the final exam is worth 20% of her math grade. What score does she need on the final to ensure her final math grade is at least 85?

Then:

Let \( x \) = final exam score.

\[
\text{Final Math Grade} = 0.8 \times 87 + 0.2 \times x = 69.6 + 0.2x
\]

Set ≥ 85:

\[
69.6 + 0.2x \geq 85
\]
\[
0.2x \geq 15.4
\]
\[
x \geq 77
\]

So she needs at least 77 on the final.

But the problem says: “If she scores 85 on her final exam…” — so maybe it's asking: if she scores 85, does she qualify?

Then:

\[
69.6 + 0.2(85) = 69.6 + 17 = 86.6 \geq 85 \Rightarrow \text{Yes}
\]

But it asks: “what score does she need…” — so likely, the “if she scores 85” is a red herring or typo.

Alternatively, maybe “current average” is not 87, but she has a lower average?

Wait — perhaps “current average” is 87, but that’s before the final, and the final is worth 20%, and she wants to know what she needs on the final to keep her grade at 85.

But 87 is already above 85, so even if she gets 0, her grade is:

\[
0.8 \times 87 + 0.2 \times 0 = 69.6 < 85
\]

So she needs to do well.

As calculated: she needs 77 on the final.

So likely, the correct interpretation is:

> Jessica needs a final math grade of at least 85. Her current average (based on 80% of the course) is 87. The final exam is worth 20%. What score does she need on the final to achieve a final grade of at least 85?

Answer: 77 or higher

But the problem says: “If she scores 85 on her final exam…” — so maybe it's asking: if she scores 85 on the final, will she qualify?

Then:

Final grade = 0.8×87 + 0.2×85 = 69.6 + 17 = 86.6 ≥ 85 → Yes.

But the question asks: “what score does she need to get on her test to qualify?” — so it's inconsistent.

Given the confusion, I suspect a typo.

But based on standard interpretation, here's the best answer:

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✔ Problem 3 – Corrected Interpretation:

Assume:
Jessica needs a final math grade of at least 85.
Her current average (before final) is 87.
The final exam is worth 20% of the math grade.
Let \( x \) = final exam score.

Then:

\[
0.8 \times 87 + 0.2x \geq 85
\]

\[
69.6 + 0.2x \geq 85
\]

\[
0.2x \geq 15.4
\]

\[
x \geq 77
\]

So she needs at least 77 on the final exam.

Inequality:
\[
0.8(87) + 0.2x \geq 85
\]

Answer: She needs to score at least 77 on the final exam.

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## ✔ Final Answers:

Problem 1:

- Inequality: \( 800 - 75x \geq 650 \)
- Solution: \( x \leq 2 \)
- Sara can withdraw money at most 2 times.

Problem 2:

- Inequality: \( 750 + 0.19m \leq 730 \)
- Solving: \( m \leq -105.26 \) → Not possible
- You cannot afford the limo; no miles can be traveled.

Problem 3:

- Assuming the intent is to find the minimum final exam score needed to achieve a math grade of at least 85, given current average of 87 and final worth 20%.
- Inequality: \( 0.8(87) + 0.2x \geq 85 \)
- Solution: \( x \geq 77 \)
- She needs to score at least 77 on the final exam.

> Note: The original wording of Problem 3 is ambiguous or contains errors. The solution assumes a reasonable correction.

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Let me know if you'd like a visual explanation or graph of any of these inequalities!