Section 2 Topic 5 Solving Inequalities Part 1

Section 2, Topic 5: Solving Inequalities – Part 1: A Comprehensive Guide

This comprehensive guide delves into the fundamentals of solving inequalities, a crucial concept in algebra and beyond. We'll cover various inequality types, solution methods, and practical applications, ensuring you gain a solid understanding of this important mathematical skill. This is Part 1, focusing on linear inequalities.

Understanding Inequalities

Before diving into solving techniques, let's clarify what inequalities are. Unlike equations, which state that two expressions are equal (=), inequalities express a relationship where one expression is greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) another.

Key Symbols:

• >: Greater than
• <: Less than
• ≥: Greater than or equal to
• ≤: Less than or equal to

Examples:

• x > 5: x is greater than 5.
• y ≤ -2: y is less than or equal to -2.
• 3z ≥ 9: 3z is greater than or equal to 9.

Solving Linear Inequalities

Linear inequalities involve variables raised to the power of one. Solving them involves isolating the variable to find the range of values that satisfy the inequality. The process is similar to solving linear equations, but with a crucial difference: when multiplying or dividing by a negative number, you must reverse the inequality sign.

Basic Steps to Solve Linear Inequalities:

1. Simplify both sides: Combine like terms on each side of the inequality.

2. Isolate the variable term: Use addition or subtraction to move the constant terms to one side of the inequality and the variable term to the other.

3. Solve for the variable: Use multiplication or division to isolate the variable. Remember to reverse the inequality sign if you multiply or divide by a negative number.

4. Graph the solution: Represent the solution set on a number line. Use an open circle (○) for > or < (meaning the endpoint is not included) and a closed circle (●) for ≥ or ≤ (meaning the endpoint is included).

5. Write the solution in interval notation: This is a concise way to represent the solution set using parentheses and brackets. Parentheses ( ) are used for open intervals (endpoints not included), while brackets [ ] are used for closed intervals (endpoints included). For example, the solution x > 2 can be written as (2, ∞).

Examples of Solving Linear Inequalities:

Example 1: 2x + 5 > 9

1. Simplify: The inequality is already simplified.

2. Isolate the variable term: Subtract 5 from both sides: 2x > 4

3. Solve for the variable: Divide both sides by 2: x > 2

4. Graph the solution: Draw a number line, place an open circle at 2, and shade to the right (since x is greater than 2).

5. Interval notation: (2, ∞)

Example 2: -3x + 7 ≤ 13

1. Simplify: The inequality is already simplified.

2. Isolate the variable term: Subtract 7 from both sides: -3x ≤ 6

3. Solve for the variable: Divide both sides by -3. Remember to reverse the inequality sign! x ≥ -2

4. Graph the solution: Draw a number line, place a closed circle at -2, and shade to the right (since x is greater than or equal to -2).

5. Interval notation: [-2, ∞)

Example 3: 5 - 2x ≥ 11

1. Simplify: The inequality is already simplified.

2. Isolate the variable term: Subtract 5 from both sides: -2x ≥ 6

3. Solve for the variable: Divide both sides by -2. Reverse the inequality sign! x ≤ -3

4. Graph the solution: Draw a number line, place a closed circle at -3, and shade to the left (since x is less than or equal to -3).

5. Interval notation: (-∞, -3]

Example 4: 4x - 3 < 2x + 7

1. Simplify: This requires combining like terms. Subtract 2x from both sides: 2x - 3 < 7

2. Isolate the variable term: Add 3 to both sides: 2x < 10

3. Solve for the variable: Divide both sides by 2: x < 5

4. Graph the solution: Draw a number line, place an open circle at 5, and shade to the left (since x is less than 5).

5. Interval notation: (-∞, 5)

Compound Inequalities

Compound inequalities involve two or more inequalities connected by "and" or "or."

"And" Inequalities: The solution must satisfy both inequalities.

"Or" Inequalities: The solution must satisfy at least one of the inequalities.

Solving Compound Inequalities:

Example 1 (And): -3 < 2x + 1 < 7

This is a shorthand way of writing two inequalities: -3 < 2x + 1 and 2x + 1 < 7. Solve each inequality separately:

• -3 < 2x + 1: Subtract 1 from both sides: -4 < 2x. Divide by 2: -2 < x
• 2x + 1 < 7: Subtract 1 from both sides: 2x < 6. Divide by 2: x < 3

The solution is the intersection of these two solutions: -2 < x < 3.

Graph: A closed circle at -2 and a closed circle at 3. The solution is shaded between -2 and 3.

Interval Notation: (-2, 3)

Example 2 (Or): x ≤ -2 or x > 5

This means the solution includes all values less than or equal to -2 or greater than 5.

Graph: A closed circle at -2 and an open circle at 5. The solution is shaded to the left of -2 and to the right of 5.

Interval Notation: (-∞, -2] ∪ (5, ∞)

Applications of Inequalities

Inequalities are used extensively in various fields:

• Physics: Describing the range of possible values for physical quantities like speed, temperature, or pressure.

• Economics: Modeling economic relationships, such as supply and demand, or profit maximization.

• Computer Science: Defining constraints in algorithms and programming.

• Engineering: Setting tolerances and limitations for design parameters.

Advanced Concepts (Brief Overview for Part 1)

While this part focuses on linear inequalities, future parts will cover:

• Quadratic Inequalities: Involving variables raised to the power of two.
• Polynomial Inequalities: Involving higher-order polynomials.
• Rational Inequalities: Involving rational expressions (fractions with variables in the numerator or denominator).
• Absolute Value Inequalities: Involving absolute value expressions.

This comprehensive guide provides a strong foundation for understanding and solving linear inequalities. Mastering these concepts is crucial for tackling more advanced inequality problems in future studies. Remember to practice regularly to build your skills and confidence! Through consistent practice and a clear understanding of the fundamental principles, solving inequalities will become second nature. In the next part, we'll explore the more complex world of quadratic and higher-order inequalities.