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Mathematics › 🔄 Grade 7 › Two-Step Inequalities: Solve Like an Equation, Then Watch the Sign
🔄 Grade 7 · Lesson 13 of 14

Two-Step Inequalities: Solve Like an Equation, Then Watch the Sign

Undo addition or subtraction first, then multiplication or division — and flip the symbol when you divide by a negative.

Grade 7Middle School
Two-Step Inequalities: Solve Like an Equation, Then Watch the Sign — illustration
💡
The big idea: A two-step inequality like 3x + 5 ≤ 20 solves exactly like a two-step equation: undo the addition or subtraction first, then undo the multiplication or division. The one new rule is what happens when that last step involves a negative number — multiplying or dividing both sides of an inequality by a negative flips its direction, because negation reverses order on the number line.
🎯 By the end, you'll be able to
  • Solve two-step inequalities of the form px + q > r (and <, ≥, ≤)
  • Apply inverse operations in the correct order to isolate the variable
  • Explain why multiplying or dividing both sides by a negative number flips the inequality symbol
  • Graph the solution set of a two-step inequality on a number line
  • Set up and solve a two-step inequality from a real-world word problem
📎 You should already know
  • One-variable inequalities and graphing (Grade 6)
  • Solving two-step equations
  • Multiplying and dividing with negative numbers

Same two steps as an equation

A two-step inequality like 3x + 5 ≤ 20 solves exactly like a two-step equation: undo the addition or subtraction first, then undo the multiplication or division, always doing the same thing to both sides.

Subtract 5 from both sides: 3x ≤ 15. Divide both sides by 3: x ≤ 5. Because 3 is positive, the symbol never changes direction — it stays ≤ the whole way through.

A budget problem

Suppose you have $50 saved, and a concert ticket site charges a flat $5 processing fee plus $9 per ticket. How many tickets t can you afford? The total cost must be at most your $50, so: 9t + 5 ≤ 50. That's a two-step inequality — and solving it tells you the largest number of tickets you can buy.

🔑 Solve it like an equation, one step at a time

Undo addition/subtraction first, then multiplication/division — the same order you use for equations. Keep the inequality symbol pointing the same way, unless your last step multiplies or divides both sides by a negative number.

⚠️ Flip the sign when you multiply or divide by a negative

Start with a true statement: 2 < 5. Multiply both sides by −1: you get −2 and −5. But −2 is actually greater than −5 (it's farther right on the number line) — so the statement −2 < −5 is false. To keep it true, the symbol must flip: −2 > −5.

Rule: whenever you multiply or divide both sides of an inequality by a negative number, reverse the direction of the inequality symbol.

✨ The graph rules don't change

Once you've solved for x, graphing works exactly as it did for one-step inequalities: open circle for > or <, closed circle for ≥ or ≤, then a ray shaded in the direction the final symbol points.

🎮 Explore an Inequality LIVE
Drag the boundary slider to move a, drag the test-point slider to move x, and press "Change relation" to cycle through >, ≥, <, ≤. Use it to check the boundary and direction after you solve a two-step inequality by hand.
📝 Worked example: Solve and graph: 3x + 5 < 20.
  1. Subtract 5 from both sides: 3x < 15.
  2. Divide both sides by 3 (positive, no flip): x < 5.
✓ <strong>x &lt; 5</strong>: open circle at 5, ray shaded to the left.
📝 Worked example: Solve and graph: -2x + 4 ≥ 10.
  1. Subtract 4 from both sides: −2x ≥ 6.
  2. Divide both sides by −2, a negative number — flip the symbol: x ≤ −3.
✓ <strong>x &le; &minus;3</strong>: closed circle at &minus;3, ray shaded to the left.
📝 Worked example: A taxi charges a $4 base fee plus $2 per mile. Maria has at most $20 to spend. Write and solve an inequality for the number of miles m she can travel.
  1. Set up the inequality: 4 + 2m ≤ 20.
  2. Subtract 4 from both sides: 2m ≤ 16.
  3. Divide both sides by 2 (positive, no flip): m ≤ 8.
✓ Maria can travel <strong>at most 8 miles</strong> (m &le; 8).

Check your understanding

1. Solve: 4x - 3 > 9.
Add 3 to both sides: 4x > 12. Divide by 4 (positive, no flip): x > 3.
2. Solve: -5x < 20.
Divide both sides by -5, a negative number, and flip the symbol: x > -4.
3. Which step in solving an inequality requires flipping the symbol?
Dividing (or multiplying) both sides by a negative number is the only operation that flips the inequality symbol.
4. A parking garage charges a $3 base fee plus $2 per hour. You have at most $15 to spend. Which inequality models the hours h you can park?
The total cost is the $3 fee plus $2 per hour, and it must be at most $15: 3 + 2h ≤ 15.
5. Solve 3 + 2h ≤ 15 for h. What is the greatest whole number of hours you can park?
Subtract 3: 2h ≤ 12. Divide by 2 (positive, no flip): h ≤ 6. The greatest whole number is 6.
✅ Key takeaways
  • A two-step inequality solves like a two-step equation: undo addition/subtraction first, then multiplication/division.
  • Adding or subtracting the same amount from both sides never flips the inequality symbol.
  • Multiplying or dividing both sides by a negative number always flips the inequality symbol.
  • After solving, graph the boundary with an open circle for > or <, a closed circle for ≥ or ≤, and shade the ray in the final symbol's direction.
  • Word problems translate into inequalities the same way equations do — the constraint's key phrase ('at most', 'at least') picks the symbol.