One-Step Inequalities | Answer Key for Teachers | Perfect for ... - Free Printable

Let’s solve each inequality step by step. We’ll isolate the variable on one side of the inequality sign, just like solving equations — but remember: if you multiply or divide by a negative number, flip the inequality sign! (None of these require that, though.)

---

1. 5 + x > 10
Subtract 5 from both sides:
x > 10 - 5 → x > 5

2. y - 3 < 8
Add 3 to both sides:
y < 8 + 3 → y < 11

3. 6a ≤ 24
Divide both sides by 6:
a ≤ 24 ÷ 6 → a ≤ 4

4. b + 7 ≥ 15
Subtract 7 from both sides:
b ≥ 15 - 7 → b ≥ 8

5. c/3 ≠ 9
Multiply both sides by 3:
c ≠ 9 × 3 → c ≠ 27

6. 11 - d ≥ 5
Subtract 11 from both sides:
-d ≥ 5 - 11 → -d ≥ -6
Now multiply both sides by -1 → flip the sign:
d ≤ 6? Wait — let’s check again.

Actually, better way:
11 - d ≥ 5
→ Subtract 11: -d ≥ -6
→ Multiply by -1 → flips inequality: d ≤ 6? But in the image it says d ≥ 6. Let me re-solve carefully.

Wait — maybe I made a mistake. Let’s do it differently:

11 - d ≥ 5
→ Move d to right and 5 to left:
11 - 5 ≥ d → 6 ≥ d → which is same as d ≤ 6

But the image shows d ≥ 6. That must be an error? Or did I misread?

Wait — look at original: “11 - d ≥ 5”

If d = 6 → 11 - 6 = 5 → satisfies ≥ 5 → OK
If d = 7 → 11 - 7 = 4 → not ≥ 5 → so d cannot be 7
So d must be ≤ 6.

But the answer shown is d ≥ 6 — that would mean d=7 gives 4, which is NOT ≥5 → wrong.

Wait — perhaps typo in problem? Or in my reading?

Looking back at user’s image description — it says for “11 - d ≥ 5” the answer is “d ≥ 6”. That can’t be right.

Let me test:
If d = 6 → 11 - 6 = 5 → OK
If d = 5 → 11 - 5 = 6 ≥ 5 → also OK
If d = 7 → 11 - 7 = 4 < 5 → not OK

So solution should be d ≤ 6.

But the worksheet says d ≥ 6 — that’s incorrect.

Wait — unless the original inequality was “11 - d ≤ 5”? Then:

11 - d ≤ 5 → -d ≤ -6 → d ≥ 6 → matches.

Ah! Probably a typo in the problem statement in the image? Or in my reading?

Looking at the user’s text: “11 - d ≥ 5” and answer “d ≥ 6” — that’s mathematically inconsistent.

But since this is a worksheet provided, and we’re to solve accurately, I must go with correct math.

However — wait, let me double-check the user’s input. In the initial prompt, under the image description, it lists:

“11 - d ≥ 5” → “d ≥ 6”

That is incorrect. Correct solution is d ≤ 6.

But perhaps I misread the operator? Let me see the image description again.

User wrote: “11 - d ≥ 5” and then “d ≥ 6”

This is a contradiction. Since I’m to solve accurately, I will solve correctly.

But to match the worksheet’s intent — maybe it’s “d - 11 ≥ 5”? No.

Another possibility: maybe it’s “11 - d ≤ 5”? Then d ≥ 6.

Given that all other answers match standard solutions, and this one doesn’t, I suspect a typo in the problem as presented.

But since the user included the image description with that pair, and I’m to solve accurately, I’ll note the discrepancy.

For now, I’ll proceed with correct math.

Actually — let’s continue with others and come back.

7. 7e > 14
Divide by 7: e > 2 → e > 2

8. f - 6 < 10
Add 6: f < 16 → f < 16

9. 7 + g ≤ 21
Subtract 7: g ≤ 14 → g ≤ 14

10. h + 3 ≠ 12
Subtract 3: h ≠ 9 → h ≠ 9

11. 2i < 14
Divide by 2: i < 7 → i < 7

12. j + 5 > 18
Subtract 5: j > 13 → j > 13

13. 8k ≥ 48
Divide by 8: k ≥ 6 → k ≥ 6

14. m/9 ≤ 4
Multiply by 9: m ≤ 36 → m ≤ 36

15. n + 6 < 15
Subtract 6: n < 9? Wait — 15 - 6 = 9 → n < 9

But the image says n < 21 — that can’t be.

n + 6 < 15 → n < 9

But worksheet says n < 21 — big discrepancy.

Wait — user wrote: “n + 6 < 15” → “n < 21”

That’s impossible. 15 - 6 = 9, not 21.

Unless it’s n + 6 < 27? Then n < 21.

Probably typo in problem.

Similarly, earlier “11 - d ≥ 5” → “d ≥ 6” is likely meant to be “11 - d ≤ 5” → d ≥ 6.

And “n + 6 < 15” → probably meant “n + 6 < 27” → n < 21.

Since this is a worksheet, and the answers are given, I think there might be typos in the problems as transcribed.

But the user said: “Solve the problem accurately.” So I must solve what’s written.

Let me list all as per accurate math:

From user’s listed problems:

1. 5 + x > 10 → x > 5 ✔
2. y - 3 < 8 → y < 11 ✔
3. 6a ≤ 24 → a ≤ 4 ✔
4. b + 7 ≥ 15 → b ≥ 8 ✔
5. c/3 ≠ 9 → c ≠ 27 ✔
6. 11 - d ≥ 5 → d ≤ 6 ✘ (worksheet says d ≥ 6)
7. 7e > 14 → e > 2 ✔
8. f - 6 < 10 → f < 16 ✔
9. 7 + g ≤ 21 → g ≤ 14 ✔
10. h + 3 ≠ 12 → h ≠ 9 ✔
11. 2i < 14 → i < 7 ✔
12. j + 5 > 18 → j > 13 ✔
13. 8k ≥ 48 → k ≥ 6 ✔
14. m/9 ≤ 4 → m ≤ 36 ✔
15. n + 6 < 15 → n < 9 ✘ (worksheet says n < 21)

So two errors in the worksheet as described.

But since the user included the image with those answers, and I’m to solve accurately, I’ll provide correct solutions.

However, for the purpose of this response, and since the instruction is to solve the problem accurately, I will output the correct mathematical solutions.

But let’s confirm once more for #6 and #15.

For #6: 11 - d ≥ 5
→ -d ≥ -6
→ d ≤ 6 (after multiplying by -1 and flipping)

For #15: n + 6 < 15
→ n < 9

Yes.

Perhaps the worksheet has different numbers? But based on user’s text, this is what’s given.

I think for accuracy, I’ll provide the correct answers.

Final Answer:

Here are the correct solutions to each inequality:

1. \( x > 5 \)
2. \( y < 11 \)
3. \( a \leq 4 \)
4. \( b \geq 8 \)
5. \( c \neq 27 \)
6. \( d \leq 6 \) *(Note: The worksheet may have a typo; correct solution is \( d \leq 6 \))*
7. \( e > 2 \)
8. \( f < 16 \)
9. \( g \leq 14 \)
10. \( h \neq 9 \)
11. \( i < 7 \)
12. \( j > 13 \)
13. \( k \geq 6 \)
14. \( m \leq 36 \)
15. \( n < 9 \) *(Note: The worksheet may have a typo; correct solution is \( n < 9 \))*

But the user might expect the worksheet's answers. However, the instruction is "solve the problem accurately", so I must go with math.

To be safe, I'll box the correct answers as per calculation.

Final Answer:
\[
\boxed{
\begin{array}{c}
x > 5 \\
y < 11 \\
a \leq 4 \\
b \geq 8 \\
c \neq 27 \\
d \leq 6 \\
e > 2 \\
f < 16 \\
g \leq 14 \\
h \neq 9 \\
i < 7 \\
j > 13 \\
k \geq 6 \\
m \leq 36 \\
n < 9 \\
\end{array}
}
\]

Note: Problems 6 and 15 appear to have typos in the original worksheet based on the given inequalities. The solutions above are mathematically correct for the stated inequalities.