How To Solve Inequalities In Interval Notation

How to Solve Inequalities in Interval Notation: A Comprehensive Guide

Inequalities, unlike equations, don't just provide a single solution; they represent a range of values that satisfy a given condition. Mastering the art of solving inequalities and expressing the solution in interval notation is crucial for success in algebra, calculus, and beyond. This comprehensive guide will equip you with the skills and understanding to tackle inequalities with confidence, translating your solutions elegantly into interval notation.

Understanding Inequalities and Interval Notation

Before diving into the solving process, let's solidify our understanding of the fundamentals. Inequalities use symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to) to compare expressions. The solution to an inequality is a set of values that make the inequality true.

Interval notation is a concise way to represent these solution sets. It uses parentheses () and brackets [] to denote whether the endpoints are included or excluded:

• Parentheses (): Indicate that the endpoint is not included in the solution set (used with < and >).
• Brackets []: Indicate that the endpoint is included in the solution set (used with ≤ and ≥).

For example:

• (2, 5) represents all numbers greater than 2 and less than 5.
• [2, 5] represents all numbers greater than or equal to 2 and less than or equal to 5.
• (2, 5] represents all numbers greater than 2 and less than or equal to 5.
• [2, 5) represents all numbers greater than or equal to 2 and less than 5.
• (-∞, 5) represents all numbers less than 5. (Negative infinity is always paired with a parenthesis).
• [5, ∞) represents all numbers greater than or equal to 5. (Positive infinity is always paired with a parenthesis).
• (-∞, ∞) represents all real numbers.

Solving Linear Inequalities

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

Example 1: Solving a Simple Linear Inequality

Solve the inequality 3x + 5 < 11 and express the solution in interval notation.

1. Subtract 5 from both sides: 3x < 6
2. Divide both sides by 3: x < 2

The solution in interval notation is (-∞, 2).

Example 2: Inequality with Negative Multiplication

Solve the inequality -2x + 4 ≥ 10 and express the solution in interval notation.

1. Subtract 4 from both sides: -2x ≥ 6
2. Divide both sides by -2 (and reverse the inequality sign): x ≤ -3

The solution in interval notation is (-∞, -3].

Solving Compound Inequalities

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

Example 3: "And" Compound Inequality

Solve the compound inequality -3 ≤ 2x + 1 < 7 and express the solution in interval notation.

This inequality means that -3 ≤ 2x + 1 and 2x + 1 < 7. We solve it by isolating x in the middle:

1. Subtract 1 from all parts: -4 ≤ 2x < 6
2. Divide all parts by 2: -2 ≤ x < 3

The solution in interval notation is [-2, 3).

Example 4: "Or" Compound Inequality

Solve the compound inequality x < -2 or x > 5 and express the solution in interval notation.

This represents two separate solution sets. The solution in interval notation is (-∞, -2) ∪ (5, ∞). The symbol ∪ denotes the union of the two sets.

Solving Quadratic Inequalities

Quadratic inequalities involve variables raised to the power of 2. Solving them requires finding the roots of the corresponding quadratic equation and testing intervals.

Example 5: Solving a Quadratic Inequality

Solve the inequality x² - 4x + 3 > 0 and express the solution in interval notation.

1. Factor the quadratic: (x - 1)(x - 3) > 0
2. Find the roots: The roots are x = 1 and x = 3. These roots divide the number line into three intervals: (-∞, 1), (1, 3), and (3, ∞).
3. Test each interval:
   • Choose a test value in (-∞, 1), such as x = 0. (0 - 1)(0 - 3) = 3 > 0. This interval satisfies the inequality.
   • Choose a test value in (1, 3), such as x = 2. (2 - 1)(2 - 3) = -1 < 0. This interval does not satisfy the inequality.
   • Choose a test value in (3, ∞), such as x = 4. (4 - 1)(4 - 3) = 3 > 0. This interval satisfies the inequality.

Therefore, the solution in interval notation is (-∞, 1) ∪ (3, ∞).

Solving Polynomial Inequalities of Higher Degree

The process for solving polynomial inequalities of higher degree is similar to that of quadratic inequalities. You find the roots (zeros) of the polynomial, and then test intervals determined by these roots.

Example 6: Higher-Degree Polynomial Inequality

Solve the inequality x³ - 4x² + 3x < 0

1. Factor the polynomial: x(x - 1)(x - 3) < 0
2. Find the roots: The roots are x = 0, x = 1, and x = 3.
3. Test intervals: Testing the intervals (-∞, 0), (0, 1), (1, 3), and (3, ∞) will reveal that the solution is (-∞, 0) ∪ (1, 3).

Solving Rational Inequalities

Rational inequalities involve fractions where the numerator or denominator (or both) contain variables. Solving these requires a slightly different approach.

Example 7: Solving a Rational Inequality

Solve the inequality (x + 2) / (x - 1) ≤ 0

1. Find critical values: The critical values are the values that make the numerator or denominator equal to zero: x = -2 and x = 1.
2. Test intervals: Test intervals (-∞, -2), (-2, 1), and (1, ∞). Remember, x = 1 is not included because it makes the denominator zero, leading to an undefined expression.
3. Determine solution: Based on the sign analysis of the rational expression, the solution will be [-2, 1).

Absolute Value Inequalities

Absolute value inequalities involve the absolute value function, denoted by | |. Remember that |x| represents the distance of x from 0.

Example 8: Solving an Absolute Value Inequality

Solve the inequality |x - 3| < 5.

This inequality means the distance between x and 3 is less than 5. This can be rewritten as a compound inequality:

-5 < x - 3 < 5

Solving this compound inequality gives -2 < x < 8. The solution in interval notation is (-2, 8).

Example 9: Solving an Absolute Value Inequality (≥)

Solve the inequality |2x + 1| ≥ 3.

This inequality represents two cases:

1. 2x + 1 ≥ 3: This gives x ≥ 1.
2. 2x + 1 ≤ -3: This gives x ≤ -2.

The solution in interval notation is (-∞, -2] ∪ [1, ∞).

Advanced Techniques and Considerations

While the examples above cover many common scenarios, solving more complex inequalities might require additional techniques, such as:

• Graphing: Graphing the functions involved can provide a visual representation of the solution set.
• Substitution: Substituting variables can simplify complex expressions.
• Numerical Methods: For inequalities that are difficult to solve analytically, numerical methods can provide approximate solutions.

Conclusion

Solving inequalities and representing the solutions in interval notation is a fundamental skill in mathematics. By understanding the rules of inequalities, mastering the different types of inequalities, and practicing the techniques outlined in this guide, you'll be able to confidently tackle a wide range of inequality problems and express your solutions accurately and concisely using interval notation. Remember to always check your work and consider using multiple approaches to ensure the accuracy of your results. Consistent practice is key to mastering this essential mathematical skill.