Solve The Inequality In Interval Notation

Solving Inequalities in Interval Notation: A Comprehensive Guide

Understanding how to solve inequalities and express the solution 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, techniques for solving them, and finally, expressing the solutions using interval notation. This will equip you with the tools to confidently tackle inequality problems in your coursework and beyond.

Introduction: What are Inequalities?

Inequalities, unlike equations, don't just represent equality; they show a relationship of greater than, less than, greater than or equal to, or less than or equal to between two expressions. These relationships are represented by the symbols:

• > (greater than)
• < (less than)
• ≥ (greater than or equal to)
• ≤ (less than or equal to)

Solving an inequality means finding the range of values that satisfy the given inequality. This solution set is often represented graphically on a number line and, importantly, using interval notation.

Understanding Interval Notation

Interval notation is a concise way to represent a set of numbers on the number line. It uses parentheses ( and ) to indicate open intervals (endpoints are not included) and square brackets [ and ] to indicate closed intervals (endpoints are included). Here's a breakdown:

• (a, b): Open interval; represents all numbers between a and b, excluding a and b.
• [a, b]: Closed interval; represents all numbers between a and b, including a and b.
• (a, b]: Represents all numbers between a and b, excluding a but including b.
• [a, b): Represents all numbers between a and b, including a but excluding b.
• (-∞, a): Represents all numbers less than a. Negative infinity (-∞) is always paired with a parenthesis.
• (a, ∞): Represents all numbers greater than a. Positive infinity (∞) is always paired with a parenthesis.
• (-∞, a]: Represents all numbers less than or equal to a.
• [a, ∞): Represents all numbers greater than or equal to a.

Solving Linear Inequalities

Linear inequalities involve only variables raised to the power of one. 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.

Steps to Solve Linear Inequalities:

1. Simplify both sides: Combine like terms on each side of the inequality.
2. Isolate the variable: Add or subtract terms to move the variable term to one side and the constant terms to the other.
3. Solve for the variable: Divide or multiply both sides by the coefficient of the variable. Remember to reverse the inequality sign if you multiply or divide by a negative number.
4. Express the solution in interval notation: Represent the solution set using the appropriate interval notation based on whether the endpoints are included or excluded.

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

1. Simplify: The inequality is already simplified.
2. Isolate the variable: 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 numbers less than 2, which is represented as (-∞, 2).

Solving Compound Inequalities

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

"And" Inequalities: The solution set includes values that satisfy both inequalities.

Example: Solve -2 ≤ 2x - 4 ≤ 6 and express the solution in interval notation.

1. Add 4 to all parts: 2 ≤ 2x ≤ 10
2. Divide all parts by 2: 1 ≤ x ≤ 5
3. Interval notation: The solution is [1, 5].

"Or" Inequalities: The solution set includes values that satisfy at least one of the inequalities.

Example: Solve x < -1 or x > 3 and express the solution in interval notation.

1. The solution is all numbers less than -1 or greater than 3.
2. Interval notation: The solution is (-∞, -1) ∪ (3, ∞). The symbol ∪ represents the union of two sets.

Solving Polynomial Inequalities

Polynomial inequalities involve polynomials of degree greater than one. Solving these requires a more systematic approach:

1. Rewrite the inequality with zero on one side: Move all terms to one side to create an inequality of the form P(x) > 0, P(x) < 0, P(x) ≥ 0, or P(x) ≤ 0, where P(x) is a polynomial.
2. Find the roots (zeros) of the polynomial: Solve the equation P(x) = 0 to find the roots.
3. Test intervals: The roots divide the number line into intervals. Test a value from each interval in the original inequality to determine whether the inequality is true or false in that interval.
4. Express the solution in interval notation: Based on the test results, express the solution set in interval notation.

Example: Solve x² - 4x + 3 > 0.

1. Factor the polynomial: (x - 1)(x - 3) > 0
2. Find the roots: x = 1 and x = 3
3. Test intervals:
   • If x < 1, (x - 1) is negative and (x - 3) is negative, so the product is positive.
   • If 1 < x < 3, (x - 1) is positive and (x - 3) is negative, so the product is negative.
   • If x > 3, (x - 1) is positive and (x - 3) is positive, so the product is positive.
4. Interval notation: The solution is (-∞, 1) ∪ (3, ∞).

Solving Rational Inequalities

Rational inequalities involve rational expressions (fractions with polynomials in the numerator and denominator). The approach is similar to solving polynomial inequalities, but with an additional step:

1. Rewrite the inequality with zero on one side: Move all terms to one side.
2. Find the critical values: These include the roots of the numerator and the roots of the denominator (where the denominator is zero).
3. Test intervals: Test a value from each interval defined by the critical values in the original inequality.
4. Express the solution in interval notation: Identify the intervals where the inequality holds true and express them using interval notation. Remember that the inequality is undefined where the denominator is zero.

Example: Solve (x + 1)/(x - 2) ≤ 0

1. Critical values: The numerator is zero at x = -1, and the denominator is zero at x = 2.
2. Test intervals:
   • x < -1: The expression is positive.
   • -1 < x < 2: The expression is negative.
   • x > 2: The expression is positive.
3. Interval notation: The solution is [-1, 2). Note that x = 2 is excluded because it makes the denominator zero.

Absolute Value Inequalities

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

Solving inequalities of the form |x| < a: This is equivalent to -a < x < a.

Example: Solve |x| < 3. The solution is -3 < x < 3, or (-3, 3).

Solving inequalities of the form |x| > a: This is equivalent to x < -a or x > a.

Example: Solve |x| > 2. The solution is x < -2 or x > 2, or (-∞, -2) ∪ (2, ∞).

Similar principles apply when dealing with more complex expressions inside the absolute value.

Frequently Asked Questions (FAQ)

Q: What happens if I multiply or divide an inequality by a negative number?

A: You must reverse the inequality sign. For example, if x < 5, then -x > -5.

Q: Can I graph the solution to an inequality on a number line?

A: Yes, graphing on a number line is a helpful visual aid to understand the solution set before converting it to interval notation. Open circles represent endpoints that are not included, while closed circles represent included endpoints.

Q: How do I handle inequalities with multiple variables?

A: Solving inequalities with multiple variables often results in a solution represented as a region in a coordinate plane, not simply an interval on a number line. This often involves graphing the inequality.

Q: What if the inequality involves an irrational expression?

A: Solving inequalities with irrational expressions often requires more advanced techniques, such as analyzing the function's behavior using derivatives or approximations.

Q: How do I check my answer?

A: Choose a value within the solution interval and plug it back into the original inequality. If the inequality holds true, your solution is likely correct. Try values outside the interval as well to confirm they do not satisfy the inequality.

Conclusion

Solving inequalities and expressing the solutions using interval notation are essential skills in mathematics. Understanding the different types of inequalities, the techniques for solving them, and the conventions of interval notation will allow you to approach inequality problems with confidence. Remember to practice regularly, working through various examples to solidify your understanding. The ability to accurately solve and represent inequalities is a cornerstone of algebraic proficiency and will serve you well in your further mathematical studies.