Master solving inequalities with multiplication and division including sign rules. Grade 9 Alberta math.
Lesson 6.5 of Linear Equations & Inequalities in Grade 9 Math: Alberta curriculum lessons.
In Lesson 6.4, you learned to solve inequalities using addition and subtraction. Now we'll do the same thing, but using multiplication and division.
Our goal is still to isolate the variable. However, there's one important new rule to watch for when negatives are involved!
With Positive Numbers. You can multiply or divide both sides of an inequality by the same positive number without changing the inequality sign.
When You Multiply or Divide by a Negative. You must reverse (flip) the inequality sign. This rule keeps the statement true!
Why Does This Happen?. Think about these two true statements:
5 > 2
Now multiply both sides by −1:
-5 > -2 (❌ NOT true!)
To fix it, flip the sign:
-5 < -2 (✓ Now it's true!)
Key Takeaways.
Up to now, you've solved one-step inequalities. You only had to undo a single operation.
Now, we'll handle multi-step inequalities, which need more than one inverse operation to isolate the variable.
These are just like multi-step equations, except we must pay attention to the inequality sign, and flip it if we ever multiply or divide by a negative number.
Key Takeaways.
Now that you know how to solve inequalities using addition, subtraction, multiplication, and division, let's apply what you've learned to real-world problems and verify that your answers make sense.
Solve: x/4 ≥ 3
Step 1: Multiply Both Sides by 4. 4 × (x/4) ≥ 3 × 4
Step 2: Simplify. x ≥ 12
Step 3: Note About the Sign. No flip needed! We multiplied by a positive number (4)
Answer: x ≥ 12. x ≥ 12
Check with x = 12:. Left side: 12 ÷ 4 = 3
Is 3 ≥ 3? Yes! ✓
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