Solve The Inequality Write The Solution In Interval Notation

Solve the Inequality: Write the Solution in Interval Notation

Solving inequalities is a crucial skill in algebra and pre-calculus, forming the foundation for many advanced mathematical concepts. Understanding how to manipulate inequalities and express solutions in interval notation is essential for success in higher-level mathematics and related fields like calculus and statistics. This comprehensive guide will walk you through various techniques for solving inequalities, focusing on expressing the final answer using interval notation.

Understanding Inequalities

Before diving into solution methods, let's clarify the fundamental differences between equations and inequalities. An equation states that two expressions are equal (=), while an inequality states that two expressions are not equal, using one of the following symbols:

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

Solving an inequality means finding the range of values for the variable that satisfy the given inequality. Unlike equations, which typically have a finite number of solutions, inequalities often have an infinite number of solutions represented by an interval.

Solving Linear Inequalities

Linear inequalities involve a variable raised to the power of 1. The solution process closely resembles 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 3x + 5 > 11.

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

The solution is x > 2. In interval notation, this is represented as (2, ∞). The parenthesis ( and ) indicate that 2 is not included in the solution set. Infinity (∞) is always represented with a parenthesis.

Example 2: Solve -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. In interval notation, this is [3, ∞). The bracket [ indicates that 3 is included in the solution set.

Solving Quadratic Inequalities

Quadratic inequalities involve a variable raised to the power of 2. Solving these inequalities requires a different approach.

Example 3: Solve x² - 4x + 3 < 0.

1. Factor the quadratic: (x - 1)(x - 3) < 0

2. Find the roots: The roots are x = 1 and x = 3.

3. Test intervals: We need to test the intervals determined by the roots: (-∞, 1), (1, 3), and (3, ∞).

   • Interval (-∞, 1): Choose a test point, say x = 0. (0 - 1)(0 - 3) = 3 > 0. This interval does not satisfy the inequality.
   • Interval (1, 3): Choose a test point, say x = 2. (2 - 1)(2 - 3) = -1 < 0. This interval satisfies the inequality.
   • Interval (3, ∞): Choose a test point, say x = 4. (4 - 1)(4 - 3) = 3 > 0. This interval does not satisfy the inequality.

Therefore, the solution is (1, 3).

Example 4: Solve x² + 2x ≥ 8.

1. Rewrite the inequality: x² + 2x - 8 ≥ 0

2. Factor the quadratic: (x + 4)(x - 2) ≥ 0

3. Find the roots: The roots are x = -4 and x = 2.

4. Test intervals:

   • Interval (-∞, -4): Choose x = -5. (-5 + 4)(-5 - 2) = 7 > 0. This interval satisfies the inequality.
   • Interval (-4, 2): Choose x = 0. (0 + 4)(0 - 2) = -8 < 0. This interval does not satisfy the inequality.
   • Interval (2, ∞): Choose x = 3. (3 + 4)(3 - 2) = 7 > 0. This interval satisfies the inequality.

The solution is (-∞, -4] ∪ [2, ∞). The symbol ∪ represents the union of the two intervals, meaning the solution includes values from both intervals. Brackets are used because the roots are included in the solution set.

Solving Polynomial Inequalities of Higher Degree

The process for solving polynomial inequalities of higher degree (degree 3 or greater) is similar to solving quadratic inequalities. You need to:

1. Find the roots of the polynomial. This may require factoring, using the rational root theorem, or numerical methods.
2. Test intervals: Test points in each interval defined by the roots to determine which intervals satisfy the inequality.

The solution will be a union of intervals.

Solving Rational Inequalities

Rational inequalities involve rational expressions (fractions). The key difference here is the need to consider the values that make the denominator zero, as these values are not part of the solution domain.

Example 5: Solve (x+1)/(x-2) > 0

1. Find the critical values: These are the values that make the numerator or denominator equal to zero. In this case, x = -1 and x = 2.
2. Test intervals:
   • (-∞, -1): Choose x = -2. (-2+1)/(-2-2) = 1/4 > 0. This interval satisfies the inequality.
   • (-1, 2): Choose x = 0. (0+1)/(0-2) = -1/2 < 0. This interval does not satisfy the inequality.
   • (2, ∞): Choose x = 3. (3+1)/(3-2) = 4 > 0. This interval satisfies the inequality.

Therefore, the solution is (-∞, -1) ∪ (2, ∞). Note that x = 2 is excluded because it would make the denominator zero.

Solving Inequalities Involving Absolute Values

Inequalities with absolute values require careful consideration of the definition of absolute value: |x| = x if x ≥ 0 and |x| = -x if x < 0.

Example 6: Solve |x - 3| < 2.

This inequality means that the distance between x and 3 is less than 2. This can be rewritten as:

-2 < x - 3 < 2

Adding 3 to all parts of the inequality:

1 < x < 5

The solution in interval notation is (1, 5).

Example 7: Solve |2x + 1| ≥ 5.

This inequality means that the distance between 2x + 1 and 0 is greater than or equal to 5. This can be rewritten as two separate inequalities:

2x + 1 ≥ 5 or 2x + 1 ≤ -5

Solving each inequality:

2x ≥ 4 => x ≥ 2

2x ≤ -6 => x ≤ -3

The solution in interval notation is (-∞, -3] ∪ [2, ∞).

Advanced Techniques and Considerations

For more complex inequalities, you might need to employ advanced techniques such as:

• Graphing: Graphing the function can visually reveal the intervals that satisfy the inequality.
• Sign charts: Sign charts systematically organize the intervals and the signs of the factors, making it easier to determine the solution.
• Numerical methods: For inequalities that are difficult to solve analytically, numerical methods can provide approximate solutions.

Interval Notation Summary

Mastering the art of solving inequalities and expressing the solutions in interval notation is a cornerstone of mathematical proficiency. This guide provides a solid foundation for tackling various types of inequalities, enabling you to confidently solve them and represent the solution sets accurately and precisely. Remember to practice regularly and to always double-check your work!