Which Values Of X Satisfy The Inequality

Which Values of x Satisfy the Inequality? A Comprehensive Guide

Solving inequalities is a crucial skill in algebra and mathematics in general. Understanding how to determine which values of x satisfy a given inequality is essential for various applications, from optimization problems to understanding the behavior of functions. This article delves deep into the process, covering various inequality types and providing detailed examples to solidify your understanding.

Understanding Inequalities

Before jumping into solving inequalities, let's establish a firm 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
• ≠: Not equal to

Unlike equations, which have specific solutions, inequalities typically have a range of solutions. This range represents all the values of the variable that make the inequality true.

Solving Linear Inequalities

Linear inequalities are inequalities involving a variable 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: Solving a simple linear inequality

Solve the inequality: 2x + 5 > 9

1. Subtract 5 from both sides: 2x > 4
2. Divide both sides by 2: x > 2

The solution to this inequality is x > 2. This means any value of x greater than 2 will satisfy the inequality.

Example 2: Involving a negative coefficient

Solve the inequality: -3x + 6 ≤ 12

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

The solution is x ≥ -2. Remember the crucial step of reversing the inequality sign!

Example 3: Inequality with Fractions

Solve the inequality: (x/2) - 3 < 1

1. Add 3 to both sides: x/2 < 4
2. Multiply both sides by 2: x < 8

The solution is x < 8.

Solving Quadratic Inequalities

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

1. Rearrange the inequality: Ensure the inequality is in the form ax² + bx + c > 0 (or < 0, ≤ 0, ≥ 0).
2. Find the roots: Solve the corresponding quadratic equation ax² + bx + c = 0. These roots will be critical points in determining the solution to the inequality.
3. Test intervals: The roots divide the number line into intervals. Choose a test value from each interval and substitute it into the inequality. If the test value satisfies the inequality, then the entire interval is part of the solution.

Example 4: Solving a quadratic inequality

Solve the inequality: x² - 4x + 3 > 0

1. Find the roots: Factor the quadratic: (x - 1)(x - 3) = 0. The roots are x = 1 and x = 3.

2. Test intervals: The roots divide the number line into three intervals: (-∞, 1), (1, 3), and (3, ∞).

   • Interval (-∞, 1): Let's test x = 0: (0)² - 4(0) + 3 = 3 > 0. This interval satisfies the inequality.
   • Interval (1, 3): Let's test x = 2: (2)² - 4(2) + 3 = -1 > 0. This is false. This interval does not satisfy the inequality.
   • Interval (3, ∞): Let's test x = 4: (4)² - 4(4) + 3 = 3 > 0. This interval satisfies the inequality.

Therefore, the solution is x < 1 or x > 3.

Solving Polynomial Inequalities of Higher Degree

The same principles applied to quadratic inequalities can be extended to polynomial inequalities of higher degrees. You'll need to find the roots of the polynomial equation and test intervals between the roots to determine which intervals satisfy the inequality. This can become more complex as the degree of the polynomial increases.

Solving Rational Inequalities

Rational inequalities involve fractions where the numerator and/or denominator contain variables. The process involves:

1. Finding the critical values: These are the values of x that make the numerator or denominator equal to zero.
2. Creating intervals: These intervals are based on the critical values, dividing the number line.
3. Testing intervals: Choose a test value from each interval and substitute it into the inequality.

Example 5: Solving a rational inequality

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

1. Critical values: x = -2 and x = 1.

2. Intervals: (-∞, -2), (-2, 1), (1, ∞)

3. Testing intervals:

   • (-∞, -2): Test x = -3: (-3 + 2) / (-3 - 1) = 1/4 > 0. This interval satisfies the inequality.
   • (-2, 1): Test x = 0: (0 + 2) / (0 - 1) = -2 > 0. This is false.
   • (1, ∞): Test x = 2: (2 + 2) / (2 - 1) = 4 > 0. This interval satisfies the inequality.

The solution is x < -2 or x > 1. Note that x = 1 is excluded because it makes the denominator zero, resulting in an undefined expression.

Graphing Inequalities

Graphing inequalities provides a visual representation of the solution set. For linear inequalities, you'll graph the line representing the equality and then shade the region that satisfies the inequality. For quadratic and other higher-degree inequalities, you'll identify the critical points and shade the appropriate intervals.

Compound Inequalities

Compound inequalities combine two or more inequalities using "and" or "or".

• "And" inequalities: The solution must satisfy both inequalities simultaneously.
• "Or" inequalities: The solution must satisfy at least one of the inequalities.

Example 6: Compound inequality with "and"

Solve the inequality: -2 < x ≤ 5

This means x must be greater than -2 AND less than or equal to 5. The solution is -2 < x ≤ 5.

Example 7: Compound inequality with "or"

Solve the inequality: x < -1 or x > 3

This means x must be less than -1 OR greater than 3. The solution is x < -1 or x > 3.

Absolute Value Inequalities

Absolute value inequalities involve the absolute value function |x|. Remember that |x| represents the distance of x from 0. Solving these requires considering both positive and negative cases.

Example 8: Absolute value inequality

Solve the inequality: |x - 2| < 3

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

-3 < x - 2 < 3

Solving for x:

-1 < x < 5

The solution is -1 < x < 5.

Applications of Inequalities

Inequalities have many real-world applications:

• Optimization problems: Finding the maximum or minimum value of a function subject to constraints.
• Engineering: Ensuring that a structure can withstand certain loads.
• Economics: Modeling supply and demand.
• Computer science: Analyzing algorithm efficiency.

Mastering the techniques for solving inequalities is a crucial step in advanced mathematical studies and practical problem-solving. By understanding the different types of inequalities and applying the appropriate methods, you can confidently tackle a wide range of problems. Remember to always check your solutions by substituting values into the original inequality to verify they satisfy the condition. Consistent practice and a firm grasp of the fundamentals will pave the way for your success in solving any inequality you encounter.