Solve The Inequality Write The Solution In Interval Notation

Solve the Inequality: A Comprehensive Guide to Interval Notation

Solving inequalities is a fundamental concept in algebra with wide-ranging applications in various fields, including calculus, statistics, and economics. Understanding how to solve inequalities and express the solution in interval notation is crucial for success in these areas. This comprehensive guide will walk you through the process, covering various types of inequalities and providing numerous examples.

Understanding Inequalities

Before diving into solving inequalities, let's establish a clear understanding of what they represent. An inequality is a mathematical statement that compares two expressions using inequality symbols:

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

Unlike equations, which have a finite number of solutions, inequalities often have an infinite number of solutions. This is because an inequality represents a range of values rather than a specific value.

Solving Linear Inequalities

Linear inequalities involve variables raised to the power of one. Solving them involves manipulating the inequality to isolate the variable, similar to solving linear equations. However, there's a 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. In interval notation, this is represented as (2, ∞). The parenthesis indicates that 2 is not included in the solution set. Infinity (∞) is always represented with a parenthesis.

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. In interval notation, this is represented as [3, ∞). The bracket indicates that 3 is included in the solution set.

Solving Compound Inequalities

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

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

This inequality means -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 is -2 < x < 3. In interval notation, this is represented as (-2, 3).

Example 4: Solve the compound inequality x < -1 OR x > 2.

This inequality means either x < -1 or x > 2. The solution is two separate intervals:

The solution is x < -1 or x > 2. In interval notation, this is represented as (-∞, -1) ∪ (2, ∞). The symbol ∪ represents the union of the two intervals.

Solving Quadratic Inequalities

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

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

1. Factor the quadratic: (x - 1)(x - 3) < 0
2. Find the roots: x = 1 and x = 3
3. Test intervals:
   • x < 1: (negative)(negative) > 0 (False)
   • 1 < x < 3: (positive)(negative) < 0 (True)
   • x > 3: (positive)(positive) > 0 (False)

The solution is 1 < x < 3. In interval notation, this is represented as (1, 3).

Example 6: Solve the inequality x² + 2x - 8 ≥ 0.

1. Factor the quadratic: (x + 4)(x - 2) ≥ 0
2. Find the roots: x = -4 and x = 2
3. Test intervals:
   • x ≤ -4: (negative)(negative) ≥ 0 (True)
   • -4 ≤ x ≤ 2: (positive)(negative) ≤ 0 (False)
   • x ≥ 2: (positive)(positive) ≥ 0 (True)

The solution is x ≤ -4 or x ≥ 2. In interval notation, this is represented as (-∞, -4] ∪ [2, ∞).

Solving Polynomial Inequalities of Higher Degree

Solving polynomial inequalities of higher degrees follows a similar approach to quadratic inequalities. You find the roots of the polynomial, test the intervals between the roots, and then express the solution in interval notation. The complexity increases with the degree of the polynomial. It often becomes advantageous to use a graphing calculator or software to visualize the polynomial and identify the intervals where the inequality holds true.

Solving Rational Inequalities

Rational inequalities involve fractions where the numerator and/or denominator contain variables. Solving them requires careful consideration of the values that make the denominator zero (which are excluded from the solution) and testing intervals.

Example 7: Solve the inequality (x - 1)/(x + 2) > 0.

1. Find the critical points: x = 1 (numerator = 0) and x = -2 (denominator = 0)
2. Test intervals:
   • x < -2: (negative)/(negative) > 0 (True)
   • -2 < x < 1: (negative)/(positive) < 0 (False)
   • x > 1: (positive)/(positive) > 0 (True)

The solution is x < -2 or x > 1. In interval notation, this is represented as (-∞, -2) ∪ (1, ∞). Remember, 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 x from zero.

Example 8: Solve the inequality |x - 3| < 2.

This inequality means -2 < x - 3 < 2.

1. Add 3 to all parts: 1 < x < 5

The solution is 1 < x < 5. In interval notation, this is represented as (1, 5).

Example 9: Solve the inequality |2x + 1| ≥ 5.

This inequality means 2x + 1 ≥ 5 OR 2x + 1 ≤ -5.

Solving the first inequality:

1. Subtract 1: 2x ≥ 4
2. Divide by 2: x ≥ 2

Solving the second inequality:

1. Subtract 1: 2x ≤ -6
2. Divide by 2: x ≤ -3

The solution is x ≤ -3 or x ≥ 2. In interval notation, this is represented as (-∞, -3] ∪ [2, ∞).

Practical Applications and Further Exploration

The ability to solve inequalities and express solutions in interval notation is crucial in numerous applications. Here are a few examples:

• Optimization problems: Finding the maximum or minimum value of a function within a specific constraint.
• Calculus: Determining intervals where a function is increasing or decreasing, or finding the domain of a function.
• Statistics: Calculating confidence intervals and determining the range of values within a certain probability.
• Economics: Modeling supply and demand, determining profit margins, and analyzing economic inequalities.

Further exploration into inequalities can include studying systems of inequalities, which involve solving multiple inequalities simultaneously, and using graphical methods to visualize and solve inequalities. Mastering these techniques will significantly enhance your mathematical skills and problem-solving capabilities across various disciplines. Remember that practice is key to building confidence and proficiency in solving inequalities. Work through numerous examples, and don't hesitate to consult additional resources for further assistance.