Solve Linear Inequalities
This tutorial explains how to solve linear inequalities step by step. You will review the fundamental properties of inequalities and apply them to solve standard inequalities, double inequalities, and inequalities containing fractional expressions.
Table of Contents
Properties of Inequalities
Let \( a \), \( b \), and \( c \) be real numbers.
- Transitive Property: If \( a < b \) and \( b < c \), then \( a < c \).
- Addition Property: If \( a < b \), then \( a + c < b + c \).
- Subtraction Property: If \( a < b \), then \( a - c < b - c \).
- Multiplication Property:
- If \( a < b \) and \( c > 0 \), then \( ca < cb \).
- If \( a < b \) and \( c < 0 \), then \( ca > cb \) (the inequality sign reverses).
These properties also hold when:
- The inequality sign is reversed.
- \( < \) is replaced by \( \le \).
- \( > \) is replaced by \( \ge \).
Key Reminder: The inequality sign reverses only when multiplying or dividing both sides by a negative number.
Examples with Solutions
Example 1: Standard Linear Inequality
Show Solution
Solve the inequality:
\[ 6x - 6 > 2x + 2 \]Solution
Add 6 to both sides:
\[ 6x > 2x + 8 \]Subtract \( 2x \) from both sides:
\[ 4x > 8 \]Divide both sides by \( 4 \):
\[ x > 2 \]Conclusion / Interval Notation:
\[ (2, +\infty) \]Matched Exercise: Solve \( 10x - 8 > 4x + 10 \)
Example 2: Inequality with Parentheses
Show Solution
Solve the inequality:
\[ 2(3x + 2) - 20 > 8(x - 3) \]Solution
Expand both sides:
\[ 6x + 4 - 20 > 8x - 24 \implies 6x - 16 > 8x - 24 \]Add 16 to both sides:
\[ 6x > 8x - 8 \]Subtract \( 8x \) from both sides:
\[ -2x > -8 \]Divide both sides by \( -2 \) and reverse the inequality sign:
\[ x < 4 \]Conclusion / Interval Notation:
\[ (-\infty, 4) \]Matched Exercise: Solve \( -3(4x + 1) + 10 > -4(x - 3) \)
Example 3: Double Inequality
Show Solution
Solve:
\[ -3 < 4(x + 2) - 3 < 9 \]Solution
Expand the expression inside:
\[ -3 < 4x + 8 - 3 < 9 \implies -3 < 4x + 5 < 9 \]Subtract 5 from all three parts:
\[ -8 < 4x < 4 \]Divide all parts by 4:
\[ -2 < x < 1 \]Conclusion / Interval Notation:
\[ (-2, 1) \]Matched Exercise: Solve \( -1 < -2(x - 3) - 3 < 7 \)
Example 4: Inequality with Fractions
Show Solution
Solve:
\[ \dfrac{x + 2}{3} - \dfrac{2}{5} < \dfrac{-x - 1}{3} - \dfrac{1}{6} \]Solution
Multiply all terms by the least common denominator \( 30 \):
\[ 10(x + 2) - 12 < 10(-x - 1) - 5 \] \[ 10x + 20 - 12 < -10x - 10 - 5 \implies 10x + 8 < -10x - 15 \]Add \( 10x \) to both sides:
\[ 20x + 8 < -15 \]Subtract 8 from both sides:
\[ 20x < -23 \]Divide both sides by 20:
\[ x < -\dfrac{23}{20} \]Conclusion / Interval Notation:
\[ \left(-\infty, -\dfrac{23}{20}\right) \]Matched Exercise: Solve \( \dfrac{x - 2}{4} - \dfrac{2}{7} < \dfrac{-x + 3}{7} - \dfrac{1}{2} \)
Answers to Matched Exercises
Exercise 1: \( 10x - 8 > 4x + 10 \)
Subtract \(4x\) and add \(8\): \(6x > 18 \implies x > 3\), i.e., \( (3, +\infty) \).
Exercise 2: \( -3(4x + 1) + 10 > -4(x - 3) \)
Expand: \( -12x - 3 + 10 > -4x + 12 \implies -12x + 7 > -4x + 12 \).
Add \(12x\) and subtract \(12\): \( -5 > 8x \implies x < -\dfrac{5}{8} \), i.e., \( \left(-\infty, -\dfrac{5}{8}\right) \).
Exercise 3: \( -1 < -2(x - 3) - 3 < 7 \)
Simplify: \( -1 < -2x + 6 - 3 < 7 \implies -1 < -2x + 3 < 7 \).
Subtract 3: \( -4 < -2x < 4 \). Divide by \(-2\) and reverse both signs: \( 2 > x > -2 \), i.e., \( -2 < x < 2 \) or \( (-2, 2) \).
Exercise 4: \( \dfrac{x - 2}{4} - \dfrac{2}{7} < \dfrac{-x + 3}{7} - \dfrac{1}{2} \)
Multiply by LCD \(28\): \( 7(x - 2) - 8 < 4(-x + 3) - 14 \implies 7x - 14 - 8 < -4x + 12 - 14 \).
Simplify: \( 7x - 22 < -4x - 2 \). Add \(4x\) and \(22\): \( 11x < 20 \implies x < \dfrac{20}{11} \), i.e., \( \left(-\infty, \dfrac{20}{11}\right) \).
Frequently Asked Questions
How do you solve a linear inequality?
To solve a linear inequality, isolate the variable using the same steps as solving a linear equation: add, subtract, multiply, or divide both sides by the same quantity. The only difference is that when you multiply or divide both sides by a negative number, you must reverse the inequality sign.
When do you reverse the inequality sign?
You reverse the inequality sign only when you multiply or divide both sides of the inequality by a negative number. For example, if \( -2x > 6 \), then dividing by \( -2 \) gives \( x < -3 \).
What is a double inequality?
A double inequality is a compound inequality of the form \( a < \text{expression} < b \). To solve it, perform the same operations on all three parts to isolate the variable in the middle.
How do you write the solution to an inequality in interval notation?
Use parentheses for strict inequalities (\( < \) or \( > \)) and square brackets for non-strict inequalities (\( \le \) or \( \ge \)). For example, \( x > 2 \) becomes \( (2, +\infty) \), and \( -2 < x < 1 \) becomes \( (-2, 1) \).
How do you solve an inequality with fractions?
To solve an inequality with fractions, multiply every term by the least common denominator (LCD) of all fractions. This eliminates the fractions and produces an equivalent inequality that is easier to solve.
More References on Inequalities
- Solve Equations, Systems of Equations, and Inequalities
- Tutorials on Inequalities
- Linear Algebra - Questions with Solutions
- Precalculus - Mathematics for Calculus - 7th Edition - James Stewart, Lothar Redlin, Saleem Watson
- Algebra and Trigonometry - 10th Edition - Michael Sullivan