Write The Solution To The Given Inequality In Interval Notation

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

Writing the solution to an inequality in interval notation is a crucial skill in algebra and beyond. It provides a concise and universally understood way to represent the range of values that satisfy a given inequality. This comprehensive guide will walk you through various types of inequalities, the techniques to solve them, and how to accurately express their solutions using interval notation. We'll cover everything from linear inequalities to those involving absolute values and quadratic expressions.

Understanding Interval Notation

Before diving into solving inequalities, let's solidify our understanding of interval notation. Interval notation uses parentheses () and brackets [] to describe a set of numbers.

• Parentheses (): Indicate that the endpoint is not included in the interval. This is used when the inequality symbol is < (less than) or > (greater than).

• Brackets []: Indicate that the endpoint is included in the interval. This is used when the inequality symbol is ≤ (less than or equal to) or ≥ (greater than or equal to).

Examples:

• (2, 5): Represents all numbers greater than 2 and less than 5. 2 and 5 are not included.
• [2, 5]: Represents all numbers greater than or equal to 2 and less than or equal to 5. 2 and 5 are included.
• (2, 5]: Represents all numbers greater than 2 and less than or equal to 5. 2 is not included, but 5 is.
• [2, ∞): Represents all numbers greater than or equal to 2 and extending to infinity. 2 is included, but infinity is always represented with a parenthesis because it's not a number.
• (-∞, 5): Represents all numbers less than 5, extending to negative infinity. 5 is not included.
• (-∞, ∞): Represents all real numbers.

Solving Linear Inequalities

Linear inequalities involve a single variable raised to the power of 1. Solving them involves manipulating the inequality to isolate the variable, similar to solving linear equations. However, a crucial difference is that when multiplying or dividing by a negative number, you must reverse the inequality sign.

Example 1: Solve 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
3. Interval notation: (2, ∞)

Example 2: Solve -2x + 7 ≤ 1 and express the solution in interval notation.

1. Subtract 7 from both sides: -2x ≤ -6
2. Divide both sides by -2 (and reverse the inequality sign): x ≥ 3
3. Interval notation: [3, ∞)

Example 3: Solve 5 - 4x < 17 and express the solution in interval notation.

1. Subtract 5 from both sides: -4x < 12
2. Divide both sides by -4 (and reverse the inequality sign): x > -3
3. Interval notation: (-3, ∞)

Solving Compound Inequalities

Compound inequalities involve two or more inequalities combined with "and" or "or".

Example 4: Solve -3 < 2x + 1 < 7 and express the solution in interval notation.

This inequality means -3 < 2x + 1 AND 2x + 1 < 7. We solve both inequalities simultaneously.

1. Subtract 1 from all parts: -4 < 2x < 6
2. Divide all parts by 2: -2 < x < 3
3. Interval notation: (-2, 3)

Example 5: Solve x + 2 ≤ -1 OR x - 3 ≥ 2 and express the solution in interval notation.

We solve each inequality separately and then combine the solutions.

1. Solve x + 2 ≤ -1: x ≤ -3
2. Solve x - 3 ≥ 2: x ≥ 5
3. Interval notation: (-∞, -3] ∪ [5, ∞) The symbol "∪" represents the union of the two intervals.

Solving Inequalities with Absolute Values

Absolute value inequalities require a slightly different approach. Remember that |x| = a means x = a or x = -a.

Example 6: Solve |x - 2| < 5 and express the solution in interval notation.

This inequality means -5 < x - 2 < 5.

1. Add 2 to all parts: -3 < x < 7
2. Interval notation: (-3, 7)

Example 7: Solve |2x + 1| ≥ 3 and express the solution in interval notation.

This inequality means 2x + 1 ≥ 3 OR 2x + 1 ≤ -3.

1. Solve 2x + 1 ≥ 3: 2x ≥ 2 => x ≥ 1
2. Solve 2x + 1 ≤ -3: 2x ≤ -4 => x ≤ -2
3. Interval notation: (-∞, -2] ∪ [1, ∞)

Solving Quadratic Inequalities

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

Example 8: Solve x² - 4x + 3 > 0 and express the solution in interval notation.

1. Factor the quadratic: (x - 1)(x - 3) > 0
2. Find the roots: x = 1 and x = 3
3. Test intervals: Consider the intervals (-∞, 1), (1, 3), and (3, ∞).
   • If x = 0 (in (-∞, 1)), the inequality is true (3 > 0).
   • If x = 2 (in (1, 3)), the inequality is false (-1 > 0).
   • If x = 4 (in (3, ∞)), the inequality is true (3 > 0).
4. Interval notation: (-∞, 1) ∪ (3, ∞)

Example 9: Solve x² + 2x - 8 ≤ 0 and express the solution in interval notation.

1. Factor the quadratic: (x + 4)(x - 2) ≤ 0
2. Find the roots: x = -4 and x = 2
3. Test intervals: Consider the intervals (-∞, -4), (-4, 2), and (2, ∞).
   • If x = -5, the inequality is true (7 ≤ 0 is false).
   • If x = 0, the inequality is true (-8 ≤ 0).
   • If x = 3, the inequality is false (7 ≤ 0 is false).
4. Interval notation: [-4, 2]

Dealing with Inequalities Involving Fractions

Inequalities involving fractions require careful consideration of the denominator. You must ensure the denominator is not zero.

Example 10: Solve (x + 1)/(x - 2) > 0 and express the solution in interval notation.

1. Find critical points: The numerator is zero when x = -1, and the denominator is zero when x = 2.
2. Test intervals: Consider the intervals (-∞, -1), (-1, 2), and (2, ∞).
   • If x = -2, the inequality is true (-1/-4 > 0).
   • If x = 0, the inequality is false (1/-2 > 0).
   • If x = 3, the inequality is true (4/1 > 0).
3. Interval notation: (-∞, -1) ∪ (2, ∞)

Remember to always check your solution by substituting values from each interval back into the original inequality to verify the results.

Advanced Techniques and Considerations

For more complex inequalities, techniques such as graphing the inequality or using a sign chart can be beneficial. These methods offer a visual representation, making it easier to identify the intervals that satisfy the inequality. Understanding the properties of inequalities, such as the transitive property (if a > b and b > c, then a > c), is crucial for solving complex problems.

This comprehensive guide provides a robust foundation for understanding and solving various types of inequalities and expressing their solutions using interval notation. Mastering these concepts is key to success in many areas of mathematics and related fields. Remember to practice regularly, tackling diverse examples to solidify your understanding and build proficiency. Consistent practice will lead to increased confidence and accuracy in handling inequalities.