Inequality Word Problems Lesson Plans & Worksheets - Free Printable

Let's solve each problem step-by-step from the 6.7 Practice Worksheet: Graphing Linear Inequalities.

---

Part 1: Tell whether the ordered pair is a solution of the linear inequality

We substitute the values of $ x $ and $ y $ into the inequality and check if the statement is true.

---

#### 1. $ x + y < -4 $; (0, 0)
Substitute $ x = 0 $, $ y = 0 $:
$ 0 + 0 = 0 $
Is $ 0 < -4 $? → No
✔ Not a solution

---

#### 2. $ y - x > -2 $; (-1, -4)
Substitute $ x = -1 $, $ y = -4 $:
$ -4 - (-1) = -4 + 1 = -3 $
Is $ -3 > -2 $? → No
✔ Not a solution

---

#### 3. $ 2x + 3y \geq 14 $; (5, 2)
Substitute $ x = 5 $, $ y = 2 $:
$ 2(5) + 3(2) = 10 + 6 = 16 $
Is $ 16 \geq 14 $? → Yes
✔ Is a solution

---

#### 4. $ x \geq -3 $; (-4, 0)
Check $ x = -4 $:
Is $ -4 \geq -3 $? → No
✔ Not a solution

---

#### 5. $ y \leq 8 $; (-9, -7)
Check $ y = -7 $:
Is $ -7 \leq 8 $? → Yes
✔ Is a solution

---

✔ Summary for Part 1:

| Problem | Is it a solution? |
|--------|-------------------|
| 1 | No |
| 2 | No |
| 3 | Yes |
| 4 | No |
| 5 | Yes |

---

Multiple Choice Questions

---

#### 6. Which ordered pair is *not* a solution of $ x + 5y < 15 $?

Test each option:

- A. (-1, -3):
$ -1 + 5(-3) = -1 -15 = -16 $
Is $ -16 < 15 $? → Yes → Solution

- B. (-1, 3):
$ -1 + 5(3) = -1 + 15 = 14 $
Is $ 14 < 15 $? → Yes → Solution

- C. (1, 3):
$ 1 + 5(3) = 1 + 15 = 16 $
Is $ 16 < 15 $? → No → Not a solution

- D. (3, 2):
$ 3 + 5(2) = 3 + 10 = 13 $
Is $ 13 < 15 $? → Yes → Solution

✔ Answer: C. (1, 3) — this is not a solution.

---

#### 7. The graph of which inequality is shown?

We are given a shaded region on a coordinate plane with a line going through points like (0, -1), (1, 0), (-1, -2), etc.

The line appears to be diagonal with slope 1 and y-intercept -1.

So the equation of the line is:
$ y = x - 1 $ or rearranged:
$ x - y = 1 $? Let's test.

Wait — let’s find the line.

From graph:
- It passes through (0, -1) → when $ x = 0 $, $ y = -1 $
- It also passes through (1, 0)

So slope $ m = \frac{0 - (-1)}{1 - 0} = 1 $

So equation: $ y = x - 1 $

Rewriting: $ x - y = 1 $? Wait:
$ y = x - 1 $ → $ x - y = 1 $? Let's check:
If $ y = x - 1 $, then $ x - y = x - (x - 1) = 1 $ → Yes! So $ x - y = 1 $

But we need an inequality.

Now look at shading: below the line? Or above?

Shaded area includes point (0, -2):
Plug into $ x - y = ? $
$ 0 - (-2) = 2 $ → Is $ 2 \geq 1 $? Yes.

But we want to match the options:

Options:

A. $ x + y \leq -1 $
B. $ x + y \geq -1 $
C. $ x - y \leq -1 $
D. $ x - y \geq -1 $

Wait — our line was $ x - y = 1 $? But that doesn't match any option directly.

Wait — maybe I made a mistake.

Let’s double-check the line.

Points on the line:
From graph:
- At $ x = 0 $, $ y = -1 $ → (0, -1)
- At $ x = 1 $, $ y = 0 $ → (1, 0)
- At $ x = -1 $, $ y = -2 $

So again, slope = 1, y-intercept = -1 → $ y = x - 1 $

Now write in standard form:
$ y = x - 1 $ → $ -x + y = -1 $ → multiply both sides by -1: $ x - y = 1 $

Still not matching options.

Wait — what if the line is $ x + y = -1 $? Let's test:

At $ x = 0 $, $ y = -1 $ → $ 0 + (-1) = -1 $ → yes
At $ x = 1 $, $ y = -2 $ → $ 1 + (-2) = -1 $ → yes
At $ x = -1 $, $ y = 0 $ → $ -1 + 0 = -1 $ → yes

So the line is $ x + y = -1 $

And the shaded region is below the line (since it includes (0, -2), which gives $ 0 + (-2) = -2 < -1 $)

So inequality is: $ x + y \leq -1 $

✔ Answer: A. $ x + y \leq -1 $

Because the shaded region is below the line $ x + y = -1 $, so $ x + y \leq -1 $

---

✔ Answer to #7: A. $ x + y \leq -1 $

---

Graph the inequalities

We’ll go over how to graph each one.

---

#### 8. $ y > x + 3 $

- Step 1: Graph the line $ y = x + 3 $ using a dashed line (because it's strict inequality: >)
- Step 2: Shade above the line (since y is greater than)
- Step 3: Pick a test point (like (0,0)):
$ 0 > 0 + 3 $? → $ 0 > 3 $? → False → So shade opposite side → Above

✔ Graph: Dashed line with slope 1, y-intercept 3, shaded above

---

#### 9. $ y < 3x + 5 $

- Step 1: Graph $ y = 3x + 5 $ as dashed line
- Step 2: Shade below the line
- Test (0,0): $ 0 < 0 + 5 $? → $ 0 < 5 $? → True → So shade below

✔ Graph: Dashed line, slope 3, y-intercept 5, shaded below

---

#### 10. $ x - y \leq -1 $

First rewrite:
$ x - y \leq -1 $ → $ -y \leq -x -1 $ → Multiply both sides by -1 (reverse inequality):
$ y \geq x + 1 $

So:
- Graph $ y = x + 1 $ as solid line (because ≤)
- Shade above the line

✔ Graph: Solid line, slope 1, y-intercept 1, shaded above

---

#### 11. $ 20x + 20 > 3y $

Solve for $ y $:

Divide both sides by 3:
$ \frac{20x + 20}{3} > y $ → $ y < \frac{20}{3}x + \frac{20}{3} $

So:
- Graph $ y = \frac{20}{3}x + \frac{20}{3} $ as dashed line (strict inequality)
- Shade below the line

Note: $ \frac{20}{3} \approx 6.67 $, so steep slope.

✔ Graph: Dashed line with slope ~6.67, y-intercept ~6.67, shaded below

---

✔ Final Answers Summary

---

Part 1: Solutions?

1. No
2. No
3. Yes
4. No
5. Yes

---

Multiple Choice:

6. C. (1, 3)
7. A. $ x + y \leq -1 $

---

Graphing Instructions:

8. $ y > x + 3 $: Dashed line, slope 1, y-int 3, shade above
9. $ y < 3x + 5 $: Dashed line, slope 3, y-int 5, shade below
10. $ x - y \leq -1 $ → $ y \geq x + 1 $: Solid line, slope 1, y-int 1, shade above
11. $ 20x + 20 > 3y $ → $ y < \frac{20}{3}x + \frac{20}{3} $: Dashed line, slope ≈6.67, y-int ≈6.67, shade below

---

Let me know if you'd like me to draw the graphs or provide a visual description!