Worksheet with multi-step inequality problems and number line graphs for solving and visualizing solutions.

It looks like you've uploaded a worksheet titled "Infinite Algebra 1: Multi-Step Inequalities", which contains 11 inequalities to solve and graph on a number line. The solutions are already filled in, but I'll go through each problem step-by-step to explain how to solve them correctly.

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Problem 1:

Inequality: $ 3 < -5x + 2x $
Simplify:
$ 3 < -3x $
Divide both sides by $-3$, remembering to reverse the inequality sign when dividing by a negative:
$ x < -1 $

✔ Answer: $ x < -1 $
Graph: Open circle at $-1$, shaded to the left.

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Problem 2:

Inequality: $ 6x + 2 + 6x \leq 14 $
Simplify:
$ 12x + 2 \leq 14 $
Subtract 2:
$ 12x \leq 12 $
Divide by 12:
$ x \leq 1 $

✔ Answer: $ x \leq 1 $
Graph: Closed circle at 1, shaded to the left.

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Problem 3:

Inequality: $ -p - 4p > -10 $
Simplify:
$ -5p > -10 $
Divide by $-5$, reverse inequality:
$ p < 2 $

✔ Answer: $ p < 2 $
Graph: Open circle at 2, shaded to the left.

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Problem 4:

Inequality: $ 18 \geq 3d + 4d $
Simplify:
$ 18 \geq 7d $
Divide by 7:
$ d \leq \frac{18}{7} \approx 2.57 $

But the answer shown is $ d \leq 2 $. That’s incorrect.

Let’s check:
- $ 7d \leq 18 $
- $ d \leq 18/7 \approx 2.57 $

So correct answer should be $ d \leq \frac{18}{7} $ or $ d \leq 2\frac{4}{7} $

✘ Given Answer: $ d \leq 2 $ → Incorrect

✔ Correct Answer: $ d \leq \frac{18}{7} $

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Problem 5:

Inequality: $ 9 + 2 - 2m \geq 2 - 3 $
Simplify:
Left: $ 11 - 2m $, Right: $ -1 $
So:
$ 11 - 2m \geq -1 $
Subtract 11:
$ -2m \geq -12 $
Divide by $-2$, reverse inequality:
$ m \leq 6 $

✔ Answer: $ m \leq 6 $
Graph: Closed circle at 6, shaded left.

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Problem 6:

Inequality: $ -3 - 4(4x + 6) > -11 $
Distribute:
$ -3 - 16x - 24 > -11 $
Simplify:
$ -16x - 27 > -11 $
Add 27:
$ -16x > 16 $
Divide by $-16$, reverse inequality:
$ x < -1 $

✔ Answer: $ x < -1 $
Graph: Open circle at $-1$, shaded left.

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Problem 7:

Inequality: $ 6 - 46n > 7(2 - 12) $
First simplify right side:
$ 7(2 - 12) = 7(-10) = -70 $
So:
$ 6 - 46n > -70 $
Subtract 6:
$ -46n > -76 $
Divide by $-46$, reverse inequality:
$ n < \frac{76}{46} = \frac{38}{23} \approx 1.65 $

But the given answer is $ n < 1 $ — this is too small.

Let’s double-check:

- $ -46n > -76 $
- Divide: $ n < 76/46 = 38/23 \approx 1.65 $

So $ n < \frac{38}{23} $

✘ Given Answer: $ n < 1 $ → Incorrect

✔ Correct Answer: $ n < \frac{38}{23} $

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Problem 8:

Inequality: $ -128 > -6(6b - 7) $
Distribute:
$ -128 > -36b + 42 $
Subtract 42:
$ -170 > -36b $
Divide by $-36$, reverse inequality:
$ b > \frac{170}{36} = \frac{85}{18} \approx 4.72 $

But the answer shown is $ b > 5 $. Let's test:

- If $ b = 5 $: $ -6(6*5 - 7) = -6(30 - 7) = -6(23) = -138 $
- Is $ -128 > -138 $? Yes.

Try $ b = 4.72 $: $ 6b - 7 = 6*(85/18) - 7 = 85/3 - 7 = (85 - 21)/3 = 64/3 ≈ 21.33 $, times $-6$: $-128$

So equality at $ b = 85/18 $

So $ b > 85/18 $

But the answer says $ b > 5 $, which is more restrictive than needed.

Let’s test $ b = 4.8 $:
$ 6*4.8 = 28.8 - 7 = 21.8 $, times $-6 = -130.8 $, and $-128 > -130.8$ → True

So $ b = 4.8 $ satisfies inequality, but $ 4.8 < 5 $

So $ b > 5 $ is not correct

✘ Given Answer: $ b > 5 $ → Incorrect

✔ Correct Answer: $ b > \frac{85}{18} \approx 4.72 $

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Problem 9:

Inequality: $ 107 \leq 6x + 7(2 - 5) $
Simplify right: $ 7(2 - 5) = 7(-3) = -21 $
So:
$ 107 \leq 6x - 21 $
Add 21:
$ 128 \leq 6x $
Divide by 6:
$ x \geq \frac{128}{6} = \frac{64}{3} \approx 21.33 $

But the answer shows $ x \geq 15 $. That’s wrong.

Check:
If $ x = 15 $: $ 6*15 = 90 $, $ 90 - 21 = 69 $, $ 107 \leq 69 $? No.

So $ x = 15 $ does not satisfy.

We need $ x \geq 64/3 \approx 21.33 $

✘ Given Answer: $ x \geq 15 $ → Incorrect

✔ Correct Answer: $ x \geq \frac{64}{3} $

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Problem 10:

Inequality: $ 5(6x + 3x) - 7 \geq 127 $
Simplify: $ 5(9x) - 7 = 45x - 7 $
So:
$ 45x - 7 \geq 127 $
Add 7:
$ 45x \geq 134 $
Divide by 45:
$ x \geq \frac{134}{45} \approx 2.98 $

The answer says $ x \geq 3 $

Check:
$ x = 3 $: $ 5(6*3 + 3*3) = 5(18 + 9) = 5*27 = 135 - 7 = 128 \geq 127 $ → True

But what about $ x = 2.98 $?
$ 9x = 26.82 $, $ 5*26.82 = 134.1 $, minus 7 = 127.1 ≥ 127 → True

So $ x \geq 134/45 \approx 2.977... $, so slightly less than 3

But since it's a fraction, and we’re dealing with real numbers, the exact answer is $ x \geq \frac{134}{45} $

But the answer given is $ x \geq 3 $, which is a little too strict.

However, if only integers are considered, then $ x \geq 3 $ would be acceptable.

But as a real-number inequality, it should be $ x \geq \frac{134}{45} $

So:

✘ Given Answer: $ x \geq 3 $ → Approximately correct, but not precise

✔ Best Answer: $ x \geq \frac{134}{45} $

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Problem 11:

Inequality: $ -8x + 2x - 36 < -3x + 7x $
Simplify both sides:
Left: $ -6x - 36 $
Right: $ 4x $
So:
$ -6x - 36 < 4x $
Add $6x$:
$ -36 < 10x $
Divide by 10:
$ x > -3.6 $

But the answer shows $ x > -2 $

Test $ x = -3 $:
Left: $ -6*(-3) - 36 = 18 - 36 = -18 $
Right: $ 4*(-3) = -12 $
Is $ -18 < -12 $? Yes → True

Now $ x = -3.5 $:
Left: $ -6*(-3.5) - 36 = 21 - 36 = -15 $
Right: $ 4*(-3.5) = -14 $
Is $ -15 < -14 $? Yes → True

Now $ x = -2 $:
Left: $ -6*(-2) - 36 = 12 - 36 = -24 $
Right: $ 4*(-2) = -8 $
Is $ -24 < -8 $? Yes

But we need to find where it starts being true.

From earlier: $ x > -3.6 $

So $ x > -3.6 $ is correct.

But the answer says $ x > -2 $, which is not correct because values between $-3.6$ and $-2$ also work.

For example, $ x = -3 $ works, but $ x = -3 $ is less than -2

So $ x > -2 $ excludes valid solutions.

✘ Given Answer: $ x > -2 $ → Incorrect

✔ Correct Answer: $ x > -3.6 $ or $ x > -\frac{18}{5} $

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Problem 12:

Inequality: $ -1 - 4x - 6 > -11 - 7x $
Simplify both sides:
Left: $ -7 - 4x $
Right: $ -11 - 7x $
So:
$ -7 - 4x > -11 - 7x $
Add $7x$:
$ -7 + 3x > -11 $
Add 7:
$ 3x > -4 $
Divide by 3:
$ x > -\frac{4}{3} \approx -1.33 $

But the answer shows $ x > -1 $

Test $ x = -1.2 $:
Left: $ -7 -4*(-1.2) = -7 + 4.8 = -2.2 $
Right: $ -11 -7*(-1.2) = -11 + 8.4 = -2.6 $
Is $ -2.2 > -2.6 $? Yes → True

But $ x = -1.2 < -1 $, so it satisfies, but $ x > -1 $ would exclude it.

So $ x > -1 $ is too restrictive

✘ Given Answer: $ x > -1 $ → Incorrect

✔ Correct Answer: $ x > -\frac{4}{3} $

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✔ Summary of Corrections:

| Problem | Given Answer | Correct Answer |
|--------|--------------|----------------|
| 4 | $ d \leq 2 $ | $ d \leq \frac{18}{7} $ |
| 7 | $ n < 1 $ | $ n < \frac{38}{23} $ |
| 8 | $ b > 5 $ | $ b > \frac{85}{18} $ |
| 9 | $ x \geq 15 $ | $ x \geq \frac{64}{3} $ |
| 10 | $ x \geq 3 $ | $ x \geq \frac{134}{45} $ |
| 11 | $ x > -2 $ | $ x > -\frac{18}{5} $ |
| 12 | $ x > -1 $ | $ x > -\frac{4}{3} $ |

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🔍 Final Notes:

- Several answers on the worksheet are incorrect.
- Common errors:
- Not simplifying expressions fully
- Incorrectly solving multi-step inequalities
- Forgetting to reverse inequality signs when dividing by negatives
- Rounding or approximating too early

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Would you like me to generate a corrected version of this worksheet with proper solutions and graphs?