Linear Inequalities

Linear inequalities are mathematical expressions involving variables of degree one, connected by inequality symbols such as \(<\), \(>\), \(\leq\), and \(\geq\). They represent a range of values rather than a single solution.

Concept Explanation

A linear inequality in one variable has the form \(ax + b < 0\), \(ax + b \leq 0\), \(ax + b > 0\), or \(ax + b \geq 0\), where \(a \neq 0\). For two variables, it takes the form \(ax + by + c < 0\), etc., where \(a\) and \(b\) are not both zero.

These inequalities define regions on the number line (one variable) or the coordinate plane (two variables).

Formula Derivation

Rules for solving linear inequalities:

• If \(a \geq b\), then \(a \pm k \geq b \pm k\) for any real number \(k\).
• If \(a \geq b\) and \(k > 0\), then \(ka \geq kb\).
• If \(a \geq b\) and \(k < 0\), then \(ka \leq kb\) (inequality sign reverses).

These rules allow manipulation of inequalities similarly to equations, with caution when multiplying or dividing by negative numbers.

Worked Illustrations

Example 1: Solve \(3x - 5 < 7\).

Step 1: Add 5 to both sides:

\[ 3x - 5 + 5 < 7 + 5 \\ 3x < 12 \]

Step 2: Divide both sides by 3 (positive number, so inequality sign remains):

\[ x < \frac{12}{3} \\ x < 4 \]

Example 2: Solve \(-2x + 3 \geq 7\).

Step 1: Subtract 3 from both sides:

\[ -2x + 3 - 3 \geq 7 - 3 \\ -2x \geq 4 \]

Step 2: Divide both sides by \(-2\) (negative number, reverse inequality):

\[ x \leq \frac{4}{-2} \\ x \leq -2 \]

Solved Examples

Example 3: Solve \(5 - 2x > 1\).

Step 1: Subtract 5 from both sides:

\[ 5 - 2x - 5 > 1 - 5 \\ -2x > -4 \]

Step 2: Divide both sides by \(-2\) (negative number, reverse inequality):

\[ x < \frac{-4}{-2} \\ x < 2 \]

Practice Set

Level 1 – Easy

• Solve \(x + 3 > 7\).
• Solve \(4x - 1 \leq 11\).
• Solve \(2x + 5 < 15\).

Level 2 – Moderate

• Solve \(-3x + 4 \geq 10\).
• Solve \(5 - 2x < 9\).
• Solve \(7x + 2 \leq 3x + 10\).

Level 3 – Challenging

• Solve \(\frac{2x - 3}{4} > 1\).
• Solve \(3(2x - 1) \leq 2(x + 5)\).
• Solve \(-4x + 7 > 3x - 8\).

Answer Key

• Level 1
  • \(x > 4\)
  • \(x \leq 3\)
  • \(x < 5\)
• Level 2
  • \(x \leq -2\)
  • \(x > -2\)
  • Subtract \(3x\) both sides: \(4x - 3x + 2 \leq 10\) \(\Rightarrow x + 2 \leq 10\) \(\Rightarrow x \leq 8\)
• Level 3
  • Multiply both sides by 4: \(2x - 3 > 4\) \(\Rightarrow 2x > 7\) \(\Rightarrow x > \frac{7}{2}\)
  • Expand: \(6x - 3 \leq 2x + 10\) \(\Rightarrow 6x - 2x \leq 10 + 3\) \(\Rightarrow 4x \leq 13\) \(\Rightarrow x \leq \frac{13}{4}\)
  • Bring terms: \(-4x - 3x > -8 - 7\) \(\Rightarrow -7x > -15\) Divide by \(-7\) (reverse inequality): \(x < \frac{15}{7}\)

Quick Reference

Operation	Effect on Inequality
Add or subtract same number	Inequality sign remains same
Multiply or divide by positive number	Inequality sign remains same
Multiply or divide by negative number	Inequality sign reverses

Glossary

• Inequality: A relation between two expressions using \(<, >, \leq, \geq\).
• Linear Inequality: Inequality involving variables of degree one.
• Solution Set: All values satisfying the inequality.
• Replacement Set: Set of values from which variable values are chosen.
• Strict Inequality: Inequalities with \(<\) or \(>\).
• Slack Inequality: Inequalities with \(\leq\) or \(\geq\).