Packets  /  Solving Linear Inequalities
Grades 7-9Algebra

Solving Linear Inequalities

Inequalities are like equations with one big twist: the sign flips when you multiply or divide by a negative number. Learn the symbols, graph solutions on a number line, solve one-step and two-step inequalities, and tackle compound and word problems.

1Inequality Symbols

Key idea: Four symbols compare two values: < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to).
  1. The < and > symbols are strict: the endpoint is NOT included.
  2. The ≤ and ≥ symbols include the endpoint itself.
  3. Memory trick: the symbol always opens toward the BIGGER number, like an alligator eating the bigger meal.
  4. Learn the clue words: 'at least' means ≥, 'at most' means ≤, 'more than' means >, 'fewer than' means <.
Worked example
x > 5
x is strictly bigger than 5. So 6 works, 100 works, but 5 itself does NOT work.
Worked example
x ≤ 4
x is 4 or smaller. So 4, 0, and -100 all work.
Watch out: Do not mix up < and ≤. x < 5 does NOT include 5, but x ≤ 5 does. One tiny bar changes the answer.

2Graphing on a Number Line

Key idea: A number-line graph shows every solution at once: open circle = endpoint NOT included, closed circle = endpoint included, and the shading points toward the numbers that work.
  1. Mark the endpoint number on the number line.
  2. Use an open circle for < or > and a closed circle for ≤ or ≥.
  3. Shade right for > and ≥ (toward bigger numbers), shade left for < and ≤ (toward smaller numbers).
  4. Test one number on the shaded side in the original inequality to confirm.
Worked example
x > 3
Mark 3 with an OPEN circle (3 is not included), then shade to the RIGHT toward bigger numbers.
Worked example
x ≤ -1
Mark -1 with a CLOSED circle (-1 is included), then shade to the LEFT toward smaller numbers.
Watch out: Shading the wrong direction is the #1 graphing mistake. Always test a number from the shaded side before you finish.

3One-Step Inequalities

Key idea: Solve one-step inequalities exactly like one-step equations: undo the addition or subtraction. Adding or subtracting never changes the inequality sign.
  1. Identify the operation happening to x (plus or minus a number).
  2. Do the inverse operation on BOTH sides.
  3. Simplify each side.
  4. Test a value from your answer in the original inequality.
Worked example
x + 8 > 15
Subtract 8 from both sides: x > 15 – 8, so x > 7. Check: x = 10 gives 10 + 8 = 18 > 15.
Worked example
x – 3 ≤ 10
Add 3 to both sides: x ≤ 10 + 3, so x ≤ 13. Check: x = 13 gives 13 – 3 = 10 ≤ 10.
Watch out: Adding or subtracting never flips the sign — positive or negative, it does not matter. Only multiplying or dividing by a negative flips it.

4Two-Step Inequalities

Key idea: Two-step inequalities follow the same order as two-step equations: undo addition and subtraction first, then undo multiplication and division.
  1. Move the constant term to the other side (add or subtract — the sign stays).
  2. Divide by the coefficient of x.
  3. If the coefficient is negative, FLIP the inequality sign.
  4. Test a value from your answer in the original.
Worked example
2x + 5 < 15
Subtract 5 first: 2x < 10. Then divide by 2 (positive, no flip): x < 5. Check: x = 0 gives 5 < 15.
Worked example
3x – 7 ≥ 14
Add 7: 3x ≥ 21. Divide by 3 (positive, no flip): x ≥ 7. Check: x = 7 gives 21 – 7 = 14.
Watch out: The flip only happens at the multiply/divide step, and only when the number is negative. Dividing by positive 3 keeps the sign.

5The Sign-Flip Rule

Key idea: When you multiply or divide both sides by a NEGATIVE number, the inequality reverses: > becomes <, < becomes >, ≤ becomes ≥, ≥ becomes ≤.
  1. Isolate x exactly like an equation until you must multiply or divide.
  2. Before dividing by a negative, flag it: the sign is about to flip.
  3. Divide (or multiply), then flip the symbol.
  4. Test a value — the check catches a forgotten flip every time.
Worked example
-4x > 12
Divide by -4 and FLIP > to <: x < 12 / (-4), so x < -3. Check: x = -4 gives -4(-4) = 16 > 12.
Worked example
5 – 2x ≤ 11
Subtract 5: -2x ≤ 6. Divide by -2 and FLIP ≤ to ≥: x ≥ -3. Check: x = -3 gives 5 + 6 = 11.
Watch out: Forgetting to flip is the most common exam error on inequalities. Why does it flip? If -4x > 12, then x must be negative enough that multiplying by -4 makes a big positive — that is why x < -3.

6Compound Inequalities

Key idea: A compound inequality joins two comparisons: a < x < b means x is between a and b (an 'and' statement). Whatever you do, do it to all three parts.
  1. Recognize the a < x < b form: x must satisfy both sides.
  2. Apply the same operation to ALL THREE parts at once.
  3. Keep the bounds in order: smaller number on the left.
  4. If you multiply or divide all parts by a negative, flip BOTH symbols — the bound order reverses too.
Worked example
-2 < 2x ≤ 10
Do the same operation to ALL three parts. Divide everything by 2 (positive, no flip): -1 < x ≤ 5.
Worked example
3 < x < 8
Read it as 'x is greater than 3 AND less than 8'. Graph: open circles at 3 and 8, shade the segment between them.
Watch out: Never flip the order of the bounds. After dividing -2 < 2x ≤ 10 by 2 you get -1 < x ≤ 5, NOT 5 ≤ x < -1. Keep small on the left.

7Inequalities in Word Problems

Key idea: Translate clue words into symbols, write the inequality, solve it, then check that the answer makes sense in the story (and whether it must be a whole number).
  1. Name the unknown (n = number of notebooks, h = hours).
  2. Translate clue words: 'at least' → ≥, 'at most' → ≤, 'more than' → >, 'fewer than' → <.
  3. Write the inequality and solve it with the usual steps.
  4. Check the answer in the story — you cannot buy 6.5 notebooks.
Worked example
Each notebook costs $2.50. You have $20 and must keep $5 for lunch. How many notebooks can you buy?
Money for notebooks: 20 – 5 = 15. So 2.50n ≤ 15. Divide: n ≤ 6. You can buy at most 6 notebooks. Check: 6 notebooks = $15, leaving exactly $5.
Worked example
The robotics team needs at least 50 volunteer hours to qualify.
'At least 50' means 50 or more: h ≥ 50.
Watch out: Watch for hidden totals: 'keep at least $5 of your $20' means the notebooks must cost at most $15. Also check whether fractional answers make sense.
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60-Second Challenge

How many solving linear inequalities problems can you solve in 60 seconds?