How To Solve Inequalities In Interval Notation

How to Solve Inequalities in Interval Notation: A Comprehensive Guide

Inequalities, unlike equations, don't just offer one solution; they present a range of solutions. Mastering how to solve inequalities is crucial for various mathematical applications, from calculus to linear programming. This comprehensive guide will take you through the process of solving inequalities and expressing the solution set using interval notation. We'll cover various types of inequalities, provide step-by-step examples, and delve into the nuances of representing solutions in interval notation.

Understanding Inequalities

Before diving into the solution process, let's refresh our understanding of inequalities. An inequality is a mathematical statement that compares two expressions using inequality symbols:

• <: less than
• >: greater than
• ≤: less than or equal to
• ≥: greater than or equal to

These symbols indicate that the expressions on either side of the inequality are not equal, but rather one is greater or less than the other (or greater than or equal to, less than or equal to).

Solving Linear Inequalities

Linear inequalities involve only linear expressions (expressions with variables raised to the power of 1). Solving them is similar to solving linear equations, with one crucial difference: multiplying or dividing by a negative number reverses the inequality sign.

Steps to Solve Linear Inequalities:

1. Simplify both sides: Combine like terms and simplify both sides of the inequality.

2. Isolate the variable: Use inverse operations (addition, subtraction, multiplication, division) to isolate the variable on one side of the inequality. Remember to perform the same operation on both sides.

3. Reverse the inequality sign (if necessary): If you multiply or divide both sides by a negative number, reverse the direction of the inequality sign.

4. Express the solution in interval notation: Once you've solved for the variable, express the solution set in interval notation.

Example 1: Solve 3x + 5 > 11

1. Simplify: The inequality is already simplified.

2. Isolate x: Subtract 5 from both sides: 3x > 6

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

4. Interval Notation: The solution is all values of x greater than 2. In interval notation, this is represented as (2, ∞). The parenthesis indicates that 2 is not included in the solution set, and ∞ (infinity) represents unboundedness.

Example 2: Solve -2x + 7 ≤ 1

1. Simplify: The inequality is already simplified.

2. Isolate x: Subtract 7 from both sides: -2x ≤ -6

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

4. Interval Notation: The solution is all values of x greater than or equal to 3. In interval notation, this is represented as [3, ∞). The square bracket indicates that 3 is included in the solution set.

Solving Quadratic Inequalities

Quadratic inequalities involve quadratic expressions (expressions with variables raised to the power of 2). Solving them requires a slightly different approach.

Steps to Solve Quadratic Inequalities:

1. Rewrite the inequality in standard form: Move all terms to one side, so the inequality is in the form ax² + bx + c < 0 (or >, ≤, ≥ 0).

2. Find the roots (zeros) of the corresponding quadratic equation: Set the quadratic expression equal to zero and solve for x. This can be done by factoring, using the quadratic formula, or completing the square.

3. Determine the intervals: The roots divide the number line into intervals.

4. Test each interval: Choose a test point within each interval and substitute it into the original inequality. If the inequality is true for the test point, then the interval is part of the solution set.

5. Express the solution in interval notation: Combine the intervals that satisfy the inequality and express the solution set using interval notation.

Example 3: Solve x² - 4x + 3 < 0

1. Standard Form: The inequality is already in standard form.

2. Find Roots: Factor the quadratic: (x - 1)(x - 3) = 0. The roots are x = 1 and x = 3.

3. Determine Intervals: The roots divide the number line into three intervals: (-∞, 1), (1, 3), and (3, ∞).

4. Test Intervals:

   • (-∞, 1): Let's test x = 0. 0² - 4(0) + 3 = 3 > 0. This interval is not part of the solution.
   • (1, 3): Let's test x = 2. 2² - 4(2) + 3 = -1 < 0. This interval is part of the solution.
   • (3, ∞): Let's test x = 4. 4² - 4(4) + 3 = 3 > 0. This interval is not part of the solution.

5. Interval Notation: The solution is the interval (1, 3).

Solving Polynomial Inequalities of Higher Degree

The approach for solving polynomial inequalities of higher degree (degree 3 or greater) is similar to that for quadratic inequalities.

1. Rewrite in standard form: Move all terms to one side.

2. Find the roots: Factor the polynomial if possible; otherwise, use numerical methods to find the roots.

3. Determine intervals: The roots divide the number line into intervals.

4. Test intervals: Select a test point in each interval and check if the inequality holds.

5. Express the solution: Write the solution in interval notation.

Solving Rational Inequalities

Rational inequalities involve rational expressions (fractions with polynomials in the numerator and denominator). Solving them requires additional care.

Steps to Solve Rational Inequalities:

1. Rewrite in standard form: Move all terms to one side, creating a single rational expression.

2. Find the critical values: These are the values that make the numerator or denominator equal to zero.

3. Determine intervals: The critical values divide the number line into intervals.

4. Test intervals: Choose a test point from each interval and substitute into the inequality.

5. Express the solution: Write the solution in interval notation. Remember that the values that make the denominator zero are excluded from the solution set.

Example 4: Solve (x + 1)/(x - 2) > 0

1. Standard Form: Already in standard form.

2. Critical Values: The critical values are x = -1 (numerator = 0) and x = 2 (denominator = 0).

3. Intervals: (-∞, -1), (-1, 2), (2, ∞)

4. Test Intervals:

   • (-∞, -1): Test x = -2. (-2+1)/(-2-2) = 1/4 > 0. This interval is part of the solution.
   • (-1, 2): Test x = 0. (0+1)/(0-2) = -1/2 < 0. This interval is not part of the solution.
   • (2, ∞): Test x = 3. (3+1)/(3-2) = 4 > 0. This interval is part of the solution.

5. Interval Notation: The solution is (-∞, -1) ∪ (2, ∞). The symbol ∪ represents the union of the two intervals.

Absolute Value Inequalities

Absolute value inequalities involve the absolute value function |x|, which represents the distance of x from 0.

Solving Absolute Value Inequalities:

The approach depends on the inequality symbol:

• |x| < a: This is equivalent to -a < x < a.
• |x| > a: This is equivalent to x < -a or x > a.

The same principles apply to more complex absolute value inequalities, often requiring the use of casework or algebraic manipulation to remove the absolute value.

Compound Inequalities

Compound inequalities involve multiple inequalities combined with "and" or "or".

• "and": The solution is the intersection of the solution sets of the individual inequalities.
• "or": The solution is the union of the solution sets of the individual inequalities.

Interval Notation: A Recap

Interval notation is a concise way to represent sets of real numbers. It uses parentheses and brackets to indicate whether the endpoints are included or excluded.

• (a, b): Open interval; a < x < b; endpoints are not included.
• [a, b]: Closed interval; a ≤ x ≤ b; endpoints are included.
• (a, b]: Half-open interval; a < x ≤ b; 'a' is not included, 'b' is included.
• [a, b): Half-open interval; a ≤ x < b; 'a' is included, 'b' is not included.
• (a, ∞): a < x; unbounded interval to positive infinity.
• [a, ∞): a ≤ x; unbounded interval to positive infinity, 'a' is included.
• (-∞, b): x < b; unbounded interval to negative infinity.
• (-∞, b]: x ≤ b; unbounded interval to negative infinity, 'b' is included.
• (-∞, ∞): All real numbers.

Mastering inequalities and interval notation is a fundamental skill in mathematics. By understanding the principles outlined in this guide and practicing with various examples, you'll be well-equipped to tackle any inequality problem you encounter. Remember to always carefully consider the inequality symbol, and double-check your work to ensure accuracy. Consistent practice is key to building confidence and proficiency in solving inequalities.