Introduction to Inequalities - Algebra I

Introduction to Inequalities

This lesson introduces inequalities and their representation. Inequalities show the relationship between expressions and how they compare to each other. Understanding inequalities is fundamental in solving real-world problems and analyzing mathematical expressions.

Outline

• What are Inequalities?
• Inequality Symbols
• Solving Simple Inequalities
• Graphing Inequalities
• Examples
• Exercises with Solutions
• Homework
• Revision Section
• Contact Us for Tutoring

What are Inequalities?

Inequalities are mathematical expressions that compare two values or expressions using inequality symbols. The most common symbols are:

• < (less than)
• > (greater than)
• ≤ (less than or equal to)
• ≥ (greater than or equal to)
• ≠ (not equal to)

For example, x > 5 means that x is greater than 5.

Examples of Inequalities

Example 1:

Consider the inequality 3x + 2 < 11. To solve for x:

1. Subtract 2 from both sides: 3x < 9
2. Divide by 3: x < 3

Example 2:

Solve the inequality 2y + 4 ≥ 12:

1. Subtract 4 from both sides: 2y ≥ 8
2. Divide by 2: y ≥ 4

Example 3:

Solve the inequality -x + 7 < 3:

1. Subtract 7 from both sides: -x < -4
2. Multiply by -1 (flip the inequality): x > 4

Example 4:

Solve the inequality 4 - 2x ≤ 6:

1. Subtract 4 from both sides: -2x ≤ 2
2. Divide by -2 (flip the inequality): x ≥ -1

Example 5:

Solve the inequality 5x - 3 > 7:

1. Add 3 to both sides: 5x > 10
2. Divide by 5: x > 2

Example 6:

Solve the inequality 6 ≤ 2x - 4:

1. Add 4 to both sides: 10 ≤ 2x
2. Divide by 2: 5 ≤ x

Exercises

Exercise 1:

Solve the inequality 7x - 2 > 12:

Exercise 2:

Solve the inequality 3 - 4y ≤ -1:

Exercise 3:

Solve the inequality -2x + 5 ≥ -3:

Exercise 4:

Solve the inequality 4 - 3x < 2:

Exercise 5:

Solve the inequality 8x + 1 ≥ 17:

Exercise 6:

Solve the inequality 5 - y < 2y + 3:

Solutions

Exercise 1: For 7x - 2 > 12: Add 2 to both sides 7x > 14, divide by 7 x > 2.

Exercise 2: For 3 - 4y ≤ -1: Subtract 3 from both sides -4y ≤ -4, divide by -4 y ≥ 1.

Exercise 3: For -2x + 5 ≥ -3: Subtract 5 from both sides -2x ≥ -8, divide by -2 x ≤ 4.

Exercise 4: For 4 - 3x < 2: Subtract 4 from both sides -3x < -2, divide by -3 x > \frac{2}{3}.

Exercise 5: For 8x + 1 ≥ 17: Subtract 1 from both sides 8x ≥ 16, divide by 8 x ≥ 2.

Exercise 6: For 5 - y < 2y + 3: Subtract 5 from both sides -y < 2y - 2, add y to both sides 0 < 3y - 2, add 2 to both sides 2 < 3y, divide by 3 \frac{2}{3} < y.

Homework

• Solve the following inequalities and graph the solutions:
  • 2x - 5 > 3
  • -3y + 4 ≤ 7
  • 5 - 2x ≥ 3
  • 4x + 1 < 9
• Write a brief explanation of how to solve inequalities and check your solution.

Revision

Review the following concepts:

• The meaning of inequality symbols and their use in expressions.
• Steps to solve simple inequalities.
• How to graph the solutions of inequalities on a number line.

Watch the Video Tutorial

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