Inequalities (Single & Multi-Step)

TipLearning Objectives

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

  • Solve single-step and multi-step inequalities.
  • Use inequality symbols correctly.
  • Apply the rule for multiplying or dividing by a negative number.
  • Represent solution sets on a number line.

Key Ideas

Inequalities work much like equations—but the solution is a range of values, not just a single number.

Inequality Symbols

  • \(<\) — less than
  • \(\le\) — less than or equal
  • \(>\) — greater than
  • \(\ge\) — greater than or equal

The Golden Rule (Most Tested)

When you multiply or divide both sides by a negative,
you must reverse the inequality symbol.

Example:
\[ -3x > 12 \]
Divide by \(-3\) (flip):
\[ x < -4 \]

Solution Sets

  • \(x > 3\) → numbers greater than 3
  • \(x \le -1\) → numbers less than or equal to -1

You can show these on a number line or in interval notation.

Common Problem Types

1. One-Step Inequalities

Example:
\[ x - 7 < 5 \]
Add 7 → \(x < 12\)

2. Inequalities with Negative Coefficients

Example:
\[ -2x \ge 10 \]
Divide by \(-2\) (flip) → \(x \le -5\)

3. Multi-Step Inequalities

Example:
\[ 3(x - 2) + 5 < 17 \]
Simplify → \(x < 6\)

4. Interval Interpretation

Example: \(x < -2\) means all values to the left of -2.

Worked Examples

Example 1 — One-Step

Solve:
\[ x - 7 < 5 \]

Solution:
Add 7 → \(x < 12\)

Example 2 — Dividing by a Negative

Solve:
\[ -2x \ge 10 \]

Divide by \(-2\) → flip:
\[ x \le -5 \]

Example 3 — Multi-Step

Solve:
\[ 3(x - 2) + 5 < 17 \]

Distribute → \(3x - 6 + 5 < 17\)
Combine → \(3x - 1 < 17\)
Add 1 → \(3x < 18\)
Divide → \(x < 6\)

WarningCommon Mistakes
  • Forgetting to flip the inequality when dividing by a negative.
  • Treating inequalities like equations (solutions are ranges, not points).
  • Combining unlike terms.
  • Not simplifying before isolating the variable.

Practice Problems

  1. \(x + 4 \ge 9\)
  2. \(-5x < 20\)
  3. \(2(x + 3) \le 10\)
  4. \(7 - 3x > 1\)
  5. \(4 - 2(3x - 1) \ge 0\)

1. \(x + 4 \ge 9\)
Subtract \(4\) from both sides:

\[ x \ge 5 \]

Answer: \(x \ge 5\)

2. \(-5x < 20\)
Divide both sides by \(-5\). Since we divide by a negative number, reverse the inequality sign:

\[ x > -4 \]

Answer: \(x > -4\)

3. \(2(x + 3) \le 10\)
Distribute the \(2\):

\[ 2x + 6 \le 10 \]

Subtract \(6\) from both sides:

\[ 2x \le 4 \]

Divide by \(2\):

\[ x \le 2 \]

Answer: \(x \le 2\)

4. \(7 - 3x > 1\)
Subtract \(7\) from both sides:

\[ -3x > -6 \]

Divide by \(-3\). Since we divide by a negative number, reverse the inequality sign:

\[ x < 2 \]

Answer: \(x < 2\)

5. \(4 - 2(3x - 1) \ge 0\)
Distribute the \(-2\):

\[ 4 - 6x + 2 \ge 0 \]

Combine like terms:

\[ 6 - 6x \ge 0 \]

Subtract \(6\) from both sides:

\[ -6x \ge -6 \]

Divide by \(-6\). Since we divide by a negative number, reverse the inequality sign:

\[ x \le 1 \]

Answer: \(x \le 1\)

Summary

  • Inequality word problems require translating words into inequality symbols.
  • Phrases such as at least and no less than mean \(\ge\), while at most and no more than mean \(\le\).
  • Define a variable before writing an inequality.
  • After solving, always interpret the solution in the context of the problem.
  • Some situations involve compound inequalities, which describe a range of acceptable values.
  • At least → \(\ge\)
  • At most → \(\le\)
  • More than → \(>\)
  • Less than → \(<\)
  • The word between usually leads to a compound inequality.
  • Check that your final answer makes sense in the real-world situation.