Solving Two-Step Inequalities

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Concept Review

Solving Two-Step Inequalities: Breaking the Balance

Imagine you're saving money for a letter: 'V', title: 'Solving Two-Step Inequalities', concept: 50 skateboard. You already have $30, and you earn $8 per hour babysitting. How many hours do you need to work to have more than enough for the skateboard? This isn't just an equation—it's an inequality, and it takes two steps to solve.

Two-step inequalities follow the form px + q > r (or with <, ≥, ≤ symbols). Just like our skateboard problem: 8h + 30 > 150, where h represents hours of babysitting.

The Two-Step Dance

Solving two-step inequalities is like undoing a wrapped present—you peel away the layers in reverse order. Let's unwrap our skateboard problem:

Example: 8h + 30 > 150

Step 1: Subtract 30 from both sides

8h + 30 - 30 > 150 - 30

8h > 120

Step 2: Divide both sides by 8

8h ÷ 8 > 120 ÷ 8

h > 15

You need to work more than 15 hours to have enough for the skateboard!

⚠️ The Flip Rule

Here's the tricky part: when you multiply or divide by a negative number, you must flip the inequality symbol!

-3x + 7 < 22

-3x < 15

x > -5 ← Notice the flip!

Think of it like looking in a mirror—everything appears backwards.

Why Two Steps?

Most real-world problems involve both addition/subtraction AND multiplication/division. Whether you're calculating phone plans (monthly fee + cost per minute), workout progress (starting weight + pounds per week), or pizza costs (delivery fee + price per topping), life rarely gives us simple one-step problems.

The key is always working backwards from the order of operations. If the original expression was built by multiplying first, then adding, you solve by subtracting first, then dividing.

🔑 Key Takeaway

Just like figuring out how many hours to work for that skateboard, two-step inequalities help us find ranges of solutions in the real world. The process is predictable: undo the addition/subtraction first, then the multiplication/division. And remember—negative multipliers flip everything around, just like our mirror analogy.

Sample questions

1. Solve for x: 2x + 3 > 9

○ x > 3

○ x < 3

○ x > 6

✓ x > 3? 2x > 6, x > 3

○ x > 3

Answer: x > 3? 2x > 6, x > 3 — Subtract 3: 2x > 6, divide by 2: x > 3.

2. Solve: 3x - 5 ≤ 10

○ x ≤ 5

○ x ≥ 5

✓ x ≤ 5? 3x ≤ 15, x ≤ 5

○ x ≤ 5

Answer: x ≤ 5? 3x ≤ 15, x ≤ 5 — Add 5: 3x ≤ 15, divide by 3: x ≤ 5.

3. Find the solution: 4x + 7 ≥ 23

○ x ≥ 4

○ x ≤ 4

○ x ≥ 4

✓ x ≥ 4? 4x ≥ 16, x ≥ 4

Answer: x ≥ 4? 4x ≥ 16, x ≥ 4 — Subtract 7: 4x ≥ 16, divide by 4: x ≥ 4.

Skills in this topic

Solve two-step inequalities of the form px + q > r
Understand when to flip the inequality symbol (multiplying/dividing by a negative)
Solve two-step inequalities with negative coefficients
Solve two-step inequalities with fractions and decimals
Determine if a specific number is a valid solution to an inequality

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