Equations and Inequalities

Solving Inequalities

Inequalities are mathematical statements that compare two expressions using inequality symbols such as <, >, ≤, or ≥. Understanding how to solve and represent inequalities is fundamental in algebra, as it allows us to describe ranges of possible values for variables.

Objectives

• Use interval notation to describe solution sets

• Apply properties of inequalities to solve algebraic inequalities

• Solve combined (compound) inequalities

Intervals and Interval Notation

Intervals are sets of real numbers that lie between two endpoints. Interval notation provides a concise way to represent these sets.

• Open Interval (a, b): All real numbers x such that a < x < b.

• Closed Interval [a, b]: All real numbers x such that a ≤ x ≤ b.

• Half-Open Intervals: (a, b] means a < x ≤ b; [a, b) means a ≤ x < b.

• Unbounded Intervals: Use ∞ or –∞ to indicate intervals that extend indefinitely in one direction.

Examples of Interval Notation

• 4 ≤ x ≤ 7: Interval notation: [4, 7]

• –3 < x < 0: Interval notation: (–3, 0)

• x > 3: Interval notation: (3, ∞)

• x ≤ 5: Interval notation: (–∞, 5]

Examples of Inequality Notation

• [0, 3): 0 ≤ x < 3

• (3, ∞): x > 3

• [3, 4]: 3 ≤ x ≤ 4

• (–∞, –5]: x ≤ –5

Properties of Inequalities

Solving inequalities often involves applying certain properties that preserve or reverse the direction of the inequality symbol.

• Nonnegative Property: For any real number a,

• Addition Property: For real numbers a, b, and c:

  • If a < b, then a + c < b + c

  • If a > b, then a + c > b + c

Example: Addition Property

• If x < –4, then x + 4 < 0

• If x > 4, then x – 4 > 0

Multiplication Property of Inequalities

• If a < b and c > 0, then ac < bc

• If a < b and c < 0, then ac > bc (the inequality reverses)

• If a > b and c > 0, then ac > bc

• If a > b and c < 0, then ac < bc

Examples: Multiplying Inequalities

• Multiplying both sides of 5 < 8 by 3: 15 < 24

• Multiplying both sides of 8 > 3 by –2: –16 < –6 (note the reversal of the inequality)

Reciprocal Property for Inequalities

• If a > 0, then

• If a < 0, then

• If b > a > 0, then

• If a < b < 0, then

Solving Inequalities: Procedures

When solving inequalities, certain algebraic manipulations preserve the direction of the inequality, while others reverse it.

• Procedures that leave the inequality unchanged:

  • Simplifying both sides

  • Adding or subtracting the same value from both sides

  • Multiplying or dividing both sides by a positive number

• Procedures that reverse the inequality:

  • Multiplying or dividing both sides by a negative number

  • Interchanging the two sides of the inequality

Examples: Solving and Graphing Inequalities

Example 7: Solving a Linear Inequality

• Solve: 5 – 4x < 9

• Subtract 5: –4x < 4

• Divide by –4 (reverse the inequality): x > –1

• Interval notation: (–1, ∞)

Example 8: Solving a Linear Inequality

• Solve: 2x – 2 ≥ 3 + x

• Simplify: x ≥ 5

• Interval notation: [5, ∞)

Example 9: Solving a Combined (Compound) Inequality

• Solve: 4 < 2x + 2 < 10

• Break into two inequalities: 4 < 2x + 2 and 2x + 2 < 10

• Solve both: 1 < x < 4

• Interval notation: (1, 4)

Example 10: Solving a Combined Inequality with Negative Multiplication

• After solving, the solution is: –6 ≤ x ≤ 0

• Interval notation: [–6, 0]

Example 11: Using the Reciprocal Property

• Solve: (2x – 4)–1 > 0

• By the reciprocal property, 2x – 4 > 0

• Solve: x > 2

• Interval notation: (2, ∞)

Creating Equivalent Inequalities

It is often useful to manipulate inequalities to create equivalent forms, especially when solving compound inequalities. For example, dividing each part of –4 < x < 0 by 2 gives –2 < x/2 < 0.

Application: Physics (Ohm’s Law)

Ohm’s Law relates voltage (E), current (I), and resistance (R) in an electric circuit: . If the voltage varies between 90 and 110 volts and resistance is 10 ohms, the current I satisfies:

• 90 ≤ 10I ≤ 110

• Divide by 10: 9 ≤ I ≤ 11

• The air conditioner draws between 9 and 11 amperes of current, inclusive.