Write The Solution To The Given Inequality In Interval Notation

Solving Inequalities and Expressing Solutions in Interval Notation: A Comprehensive Guide

Understanding how to solve inequalities and express their solutions in interval notation is a crucial skill in algebra and beyond. This comprehensive guide will walk you through the process, covering various types of inequalities and providing ample examples to solidify your understanding. We will delve into the nuances of interval notation, focusing on its advantages and practical applications. By the end, you'll be confident in tackling inequality problems and expressing your answers accurately and concisely.

Introduction to Inequalities

Unlike equations, which state that two expressions are equal, inequalities show a relationship of inequality between two expressions. The most common inequality symbols are:

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

These symbols are used to compare numerical values or algebraic expressions. Solving an inequality means finding the range of values for the variable that make the inequality true. This solution set can be represented graphically on a number line or algebraically using interval notation.

Solving Linear Inequalities

Linear inequalities involve a single variable raised to the power of one. Solving them follows a similar process to solving linear equations, with one crucial difference: when multiplying or dividing both sides of an inequality by a negative number, you must reverse the inequality sign.

Example 1: Solve the inequality 3x + 5 > 11.

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

The solution is x > 2. This means any value of x greater than 2 satisfies the inequality.

Example 2: Solve the inequality -2x + 7 ≤ 1.

1. Subtract 7 from both sides: -2x ≤ -6
2. Divide both sides by -2 and reverse the inequality sign: x ≥ 3

The solution is x ≥ 3. Notice how the inequality sign flipped because we divided by a negative number.

Solving Compound Inequalities

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

Example 3: Solve the compound inequality -3 < 2x + 1 < 7.

This inequality means that 2x + 1 is greater than -3 and less than 7. We solve it by performing the same operation on all three parts of the inequality:

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

The solution is -2 < x < 3.

Example 4: Solve the compound inequality x < -2 or x > 5.

This inequality means x is less than -2 or greater than 5. The solution is two separate intervals.

Solving Quadratic Inequalities

Quadratic inequalities involve a variable raised to the power of two. Solving these requires a different approach.

Example 5: Solve the inequality x² - 4x + 3 < 0.

1. Factor the quadratic expression: (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:
   • If x = 0 (in (-∞, 1)), then (0-1)(0-3) = 3 > 0. This interval is not part of the solution.
   • If x = 2 (in (1, 3)), then (2-1)(2-3) = -1 < 0. This interval is part of the solution.
   • If x = 4 (in (3, ∞)), then (4-1)(4-3) = 3 > 0. This interval is not part of the solution.

Therefore, the solution is 1 < x < 3.

Introduction to Interval Notation

Interval notation provides a concise way to represent solution sets of inequalities. It uses parentheses and brackets to indicate whether the endpoints are included or excluded.

• Parentheses ( ): Used when the endpoint is not included (strict inequalities < and >).
• Brackets [ ]: Used when the endpoint is included (inequalities ≤ and ≥).
• ∞ (infinity) and -∞ (negative infinity): Always used with parentheses because infinity is not a number.

Example 6: The inequality x > 2 is represented in interval notation as (2, ∞).

Example 7: The inequality x ≥ 3 is represented in interval notation as [3, ∞).

Example 8: The inequality -2 < x < 3 is represented in interval notation as (-2, 3).

Example 9: The inequality x ≤ -1 or x > 4 is represented in interval notation as (-∞, -1] ∪ (4, ∞). The symbol ∪ represents the union of two sets.

Solving Polynomial Inequalities of Higher Degree

Solving polynomial inequalities of higher degree follows a similar process to solving quadratic inequalities.

Example 10: Solve the inequality x³ - 4x² - 11x + 30 ≥ 0.

1. Find the roots: By factoring or using numerical methods, we find that the roots are x = -3, x = 2, and x = 5.
2. Test intervals: We test the intervals (-∞, -3), (-3, 2), (2, 5), and (5, ∞).
3. Determine solution intervals: After testing the intervals, we find that the inequality is satisfied in the intervals [-3, 2] and [5, ∞).
4. Express in interval notation: The solution in interval notation is [-3, 2] ∪ [5, ∞).

Solving Rational Inequalities

Rational inequalities involve rational expressions (fractions). Solving these requires careful consideration of the denominator.

Example 11: Solve the inequality (x + 2)/(x - 1) > 0.

1. Find critical points: The critical points are where the numerator is zero (x = -2) and where the denominator is zero (x = 1).
2. Test intervals: Test intervals (-∞, -2), (-2, 1), and (1, ∞).
3. Determine solution intervals: The inequality is positive in the intervals (-∞, -2) and (1, ∞).
4. Express in interval notation: The solution is (-∞, -2) ∪ (1, ∞). Note that x = 1 is excluded because it would make the denominator zero.

Absolute Value Inequalities

Absolute value inequalities involve the absolute value function |x|. Recall that |x| = x if x ≥ 0 and |x| = -x if x < 0.

Example 12: Solve the inequality |x - 3| < 2.

This means that the distance between x and 3 is less than 2. This is equivalent to -2 < x - 3 < 2. Solving this compound inequality gives 1 < x < 5. In interval notation, this is (1, 5).

Example 13: Solve the inequality |x + 1| ≥ 4.

This means that the distance between x and -1 is greater than or equal to 4. This is equivalent to x + 1 ≥ 4 or x + 1 ≤ -4. Solving these gives x ≥ 3 or x ≤ -5. In interval notation, this is (-∞, -5] ∪ [3, ∞).

Frequently Asked Questions (FAQ)

Q1: What is the difference between using parentheses and brackets in interval notation?

A1: Parentheses () indicate that the endpoint is not included in the interval (strict inequalities < and >). Brackets [] indicate that the endpoint is included (inequalities ≤ and ≥).

Q2: How do I handle inequalities with infinity?

A2: Infinity (∞) and negative infinity (-∞) are always represented with parentheses because they are not actual numbers; they represent unbounded intervals.

Q3: What if I have a compound inequality with "or"?

A3: If the compound inequality uses "or," the solution is the union of the solution sets of the individual inequalities. Use the ∪ symbol to denote the union.

Q4: Can I always factor a polynomial inequality to find the roots?

A4: Not always. For higher-degree polynomials, numerical methods might be needed to approximate the roots.

Q5: How do I check my solution to an inequality?

A5: Choose a value within each interval defined by the critical points and substitute it into the original inequality. If the inequality is true, that interval is part of the solution.

Conclusion

Solving inequalities and expressing solutions in interval notation is a fundamental skill in mathematics. By understanding the rules for manipulating inequalities, mastering the techniques for different types of inequalities, and applying the concise language of interval notation, you will be well-equipped to handle a wide range of problems in algebra and beyond. Remember to practice regularly to solidify your understanding and build confidence in your problem-solving abilities. The key is to break down complex inequalities into manageable steps, carefully consider the properties of inequalities, and systematically test solutions to ensure accuracy. With consistent practice, you'll become proficient in this essential mathematical skill.