K-12 Math (US): 6-8 Algebra K-12 Math Inequalities Solve and graph inequalities

Study Guide: Inequalities — Solve and Graph (Grade 6–8, Algebra)

1. The Driving Question

You’re saving up for a $120 concert ticket, and you earn $15 every time you mow a neighbor’s lawn. How many lawns do you have to mow to afford the ticket—and why does the answer look like a math sentence with a weird squiggly symbol instead of an equals sign? What does that symbol actually tell you about the numbers you can plug in?

2. The Core Idea — Built, Not Listed

Imagine you’re at a carnival, and you have exactly $20 to spend on rides. Each ride costs $3. You could ride 6 times ($18) and have $2 left, but you can’t ride 7 times ($21) because you’d be $1 short. The inequality 3x ≤ 20 captures this: x is the number of rides, and the "≤" says you can ride up to 6 times, but not more. Unlike an equation (which is like a balance scale with equal weights on both sides), an inequality is like a seesaw that can tilt—one side can be heavier (greater) or lighter (less) than the other. Solving it means finding all the numbers that keep the seesaw tilted the right way.

When you graph an inequality, you’re drawing a map of all the possible answers. For x ≤ 6, you’d shade a number line from 6 backward, with a closed circle at 6 (because 6 is allowed). If the inequality were x < 6, you’d use an open circle (6 isn’t allowed). The shading shows every number that works—like a spotlight on all the rides you can afford.

Key Vocabulary:
- Inequality: A math sentence that compares two expressions using symbols like <, >, ≤, or ≥.
Example: The speed limit is 55 mph, so your speed s must satisfy s ≤ 55.
Note (Grades 9–12): In calculus, inequalities define intervals (e.g., x > 2) where functions behave a certain way.

• Solution Set: All the numbers that make an inequality true.
  Example: For y + 4 > 7, the solution set is y > 3—every number bigger than 3 works.
  Note: In algebra II, solution sets can include compound inequalities (e.g., −2 < x ≤ 5).

• Closed/Open Circle: A dot on a number line that shows whether the endpoint is included (closed, for ≤ or ≥) or excluded (open, for < or >).
  Example: If a roller coaster requires riders to be at least 48 inches tall, the inequality is h ≥ 48, and the graph has a closed circle at 48.

• Shading: The part of the number line that represents all solutions to an inequality.
  Example: For x < −1, you shade everything to the left of −1, like a trail of breadcrumbs leading to all the numbers smaller than −1.

3. Assessment Translation

How This Appears on State Tests (Grades 6–8):
- Multiple Choice: Questions often show an inequality (e.g., −2x + 5 > 11) and ask which graph matches the solution. Distractors might: - Flip the inequality sign (e.g., shading the wrong direction).
- Use an open circle when it should be closed (or vice versa).
- Include extra numbers in the solution set (e.g., shading x ≤ 4 when the answer is x < 4).
- Short Answer: You might be asked to solve an inequality (e.g., 3(x − 2) ≤ 12) and graph the solution. Full credit requires: - Correctly solving for x (including flipping the inequality sign if you multiply/divide by a negative).
- Drawing the number line with the correct circle and shading.
- Evidence-Based Writing: Rare, but possible—e.g., "Explain why the solution to −x > 4 is x < −4 and not x > −4."

SAT/ACT Note (Grades 9–12):
- Inequalities appear in word problems (e.g., "A taxi charges $3 plus $0.50 per mile. If you have $15, what’s the maximum number of miles you can ride?"). The SAT often tests compound inequalities (e.g., −3 ≤ 2x + 1 < 7).

Model Proficient Response:
Prompt: Solve and graph the inequality −4x − 3 ≥ 5.
Student Response: 1. Add 3 to both sides: −4x ≥ 8.
2. Divide by −4 (flip the sign): x ≤ −2.
3. Graph: Closed circle at −2, shade left.

Why It’s Proficient: - Shows all steps, including the critical sign flip.
- Graph is accurate (closed circle, correct shading).
- No arithmetic errors.

4. Mistake Taxonomy

Mistake 1: Forgetting to Flip the Inequality Sign
Prompt: Solve −2x > 10.
Common Wrong Answer: x > −5.
Why It Loses Credit: The student divided by −2 but didn’t flip the > to <. The rule is: When you multiply or divide by a negative number, reverse the inequality sign.
Correct Approach: 1. Divide both sides by −2: x < −5.
2. Graph: Open circle at −5, shade left.

Mistake 2: Misinterpreting "At Least" or "No More Than"
Prompt: "You need at least 70 points to pass a test. Write an inequality for your score s." Common Wrong Answer: s < 70.
Why It Loses Credit: "At least" means 70 or more, so the inequality should be s ≥ 70. The student confused "at least" with "less than." Correct Approach: - "At least" → ≥.
- "No more than" → ≤.
- "Fewer than" → <.

Mistake 3: Graphing the Wrong Direction
Prompt: Graph x > 1.
Common Wrong Answer: Shading to the left of 1 (or using a closed circle).
Why It Loses Credit: The student either: - Mixed up > (shade right) and < (shade left), or - Used a closed circle (only for ≥ or ≤).
Correct Approach: 1. Open circle at 1 (because 1 is not included).
2. Shade right (all numbers greater than 1).

5. Connection Layer

• Within Math: Inequalities → Systems of Inequalities — If you graph two inequalities (e.g., y > x + 1 and y < −2x + 4), the overlapping shaded region shows all the solutions that work for both. This is how businesses decide how many of two products to make to maximize profit.

• Across Subjects: Inequalities → Physics (Force Diagrams) — In physics, inequalities describe ranges of possible forces (e.g., F ≥ 50 N means the force must be at least 50 newtons to move an object). The "≥" is like a threshold—below it, nothing happens; above it, the object accelerates.

• Outside School: Inequalities → Video Game Design — Game developers use inequalities to set boundaries (e.g., player_health > 0 means the player is alive). If your health drops to 0, the game ends—just like how x ≤ 0 means the solution set stops at 0.

6. The Stretch Question

If you solve −x > 5 and get x < −5, what happens if you plug in x = −5.0001? Does it work? What about x = −5? Why does the inequality seem to "break" at exactly −5, even though −5.0001 is technically less than −5?

Pointer Toward the Answer: The inequality x < −5 excludes −5 itself because −(−5) = 5, and 5 > 5 is false (the original inequality was −x > 5, not ≥). But −5.0001 is just barely less than −5, so −(−5.0001) = 5.0001, which is greater than 5. This shows how inequalities define boundaries—the exact point where the rule changes. In calculus, this idea becomes limits (e.g., how close can x get to −5 before the inequality stops working?).