Inequalities and Word Problem Setup

Lesson 2 of 2

In this lesson:

• Solve inequalities and graph solutions on a number line
• Apply the flip rule for negative coefficients
• Write and solve equations and inequalities from word problems

What You Will Be Able to Do

By the end of this lesson, you should be able to:

1. Write an inequality of the form or for a real-world situation
2. Solve inequalities and graph the solution on a number line
3. Use open and closed circles correctly for strict vs. non-strict inequalities
4. Recognize when to flip the inequality symbol
5. Translate a word problem into an equation or inequality and interpret the solution

Why Inequalities? A Real Context

You need at least $50 to buy a game.

• You have $12 saved and earn $8 per day mowing lawns.
• How many days until you can buy the game?

What would the inequality look like?

Inequality Symbols: A Quick Review

Number Line: Open vs. Closed Circle

• Open circle ○ — endpoint not included ( or )
• Closed circle ● — endpoint included ( or )
• Arrow points in the direction of the solution set

Example 1: Solving Without the Flip

Solve and graph:

Step 1: Subtract 4 from both sides

Step 2: Divide both sides by 3

Graph: Open circle at 3, arrow pointing left

Verify: : ✓ | : ✗

Quick Check: Open or Closed Circle?

For the solution :

Is in the solution set?

A) Yes — closed circle at 3

B) No — open circle at 3

Think about the meaning of "strictly less than."

The Flip Rule: When to Reverse the Symbol

Dividing or multiplying by a negative reverses the inequality symbol.

Consider: is true. Multiply both sides by :

• Without flipping: — false
• With flipping: — true ✓

Positive division: keep the symbol. Negative division: flip the symbol.

Inequality Example Requiring the Flip

Solve and graph:

Step 1: Subtract 1 from both sides

Step 2: Divide by — flip the symbol

Graph: Closed circle at , arrow pointing left

Verify: : ✓ | : ✗

Example 3: Verify Both In and Out

Solve:

Verify — value in solution set: : ✓ and ✓

Verify — value outside: : ✗ and ✗

Both test values confirm the solution is .

Guided Practice: Identify the Flip Step

Solve:

Rewrite as:

Your turn: Divide by .

• What is the solution?
• Did the symbol flip? Why or why not?
• Graph your solution on the number line.

Write your answer, then verify with one test value.

Misconception Check: Forgot to Flip?

Someone solved and wrote — correct?

Test : ✓ — but is false.

incorrectly excludes . Flip required: .

Your Turn: Solve Four Inequalities

Solve, identify whether a flip is needed, and graph each solution.

Pause and solve. Verify each with one test value.

Answers: Check Your Four Inequality Solutions

1. → → (closed ●, arrow left)
2. → → (flip! open ○, arrow left)
3. → → (closed ●, arrow right)
4. → → (flip! open ○, arrow right)

Problems 2 and 4 both required the flip — both had negative coefficients.

From Solving to Setting Up

The harder skill: writing the equation from context.

4-step framework for every word problem:

1. Define — Let = (specific quantity in words)
2. Write — Build the equation or inequality
3. Solve — Apply inverse operations
4. Interpret — State the answer in context

The 4-Step Framework in Action

Use these four steps on every word problem — refer back as needed.

Word Problem 1: Plumber Bill

A plumber charges $45 per hour plus a $90 service fee. The bill was $247.50. How many hours did the job take?

Define: Let = number of hours worked

Write:

Solve:

Interpret: The job took 3.5 hours.

Word Problem 2: Pizza Split

Four friends split the cost of a pizza equally, plus each paid $1.25 for a drink. Each person paid $5.50 total. What was the pizza's price?

Define: Let = each person's share of the pizza

Write:

Solve:

Interpret: Each person's pizza share was $4.25; pizza total = $17.00.

Word Problem 3: Store Deal (Inequality)

A store charges $4 per item plus a $6 membership. You have $30 to spend. How many items can you buy?

Define: Let = number of items

Write:

Solve:

Interpret: You can buy at most 6 items. (Since must be a whole number, )

Guided Practice: Write the Equation

Cell phone plan: $15/month + $0.05/text. Bill was $22.50. How many texts?

Step 1: Define — Let = number of texts

Step 2: Write — the equation is: ?

Write the equation, then solve and interpret.

Quick Check: Interpreting

The store deal solution was . What does this mean?

A) Exactly 6 items must be purchased

B) You can buy 0 to 6 items

C) You need at least 6 items

D) More than 6 items are allowed

Language Reference: "At Least" vs. "At Most"

Your Turn: Three Word Problems

Apply all 4 steps for each.

1. $50/week + $3/sale. Wants ≥ $100. How many sales?

2. Perimeter = 54 cm, length = 6 cm. Find the width.

3. At most $40; apps cost $3.50 each + $5 fee. How many apps?

Answers: Check Your Word Problem Solutions

1. → → → at least 17 sales

2. → → → width is 21 cm

3. → → → at most 10 apps

For problem 1: since sales must be whole numbers, round up to 17.

Key Takeaways: Inequalities and Word Problems

✓ Same inverse operations as equations

✓ Dividing by a negative flips the symbol

✓ Open circle for /; closed for /

Forgot to flip? Test a value from the excluded region

Solutions are ranges, not single points

Preview: Equations in Geometry and Proportions

These skills apply immediately and beyond:

• Geometry: perimeter, area, and angle equations
• Proportions: connecting to 7.RP.A ratios and rates
• 8th grade: more complex linear equations
• Graphing: inequality solution sets on coordinate planes