Solving Quadratic Inequalities Step-by-Step Guide
In mathematics, a quadratic inequality is a mathematical statement that compares a quadratic expression to a value, typically zero, using inequality symbols such as <, >, ≤, or ≥. Understanding how to solve these inequalities is a fundamental skill in algebra and calculus. Quadratic inequalities appear in various applications, from physics and engineering to economics and computer science. Mastering the techniques to solve them not only enhances your mathematical toolkit but also provides a solid foundation for more advanced topics.
This guide will delve into solving several quadratic inequalities step-by-step, providing clear explanations and strategies to tackle different scenarios. We will focus on the following inequalities:
- x² + 2x - 24 < 0
- x² + 5x - 14 < 0
- x² - 2x - 24 < 0
- x² - 9x + 20 < 0
Each of these inequalities presents a unique case, and by exploring their solutions, you will gain a comprehensive understanding of how to approach quadratic inequalities in general. We will cover the essential steps, including factoring, finding critical points, and determining the intervals that satisfy the inequality. So, let's embark on this journey to master quadratic inequalities and enhance your problem-solving skills.
Let's begin by tackling the first quadratic inequality: x² + 2x - 24 < 0. To solve this, we need to find the range of x values that make the inequality true. The primary strategy involves several key steps, including factoring the quadratic expression, identifying critical points, and testing intervals.
Step 1: Factor the Quadratic Expression
The first step is to factor the quadratic expression x² + 2x - 24. We are looking for two numbers that multiply to -24 and add to 2. These numbers are 6 and -4. Therefore, we can factor the quadratic as follows:
x² + 2x - 24 = (x + 6)(x - 4)
Factoring is a fundamental technique in algebra, and it's crucial to understand how to apply it correctly. In this case, breaking down the quadratic expression into its factors allows us to easily identify the critical points where the expression equals zero.
Step 2: Find the Critical Points
The critical points are the values of x that make the factored expression equal to zero. These points are found by setting each factor equal to zero and solving for x:
x + 6 = 0 => x = -6 x - 4 = 0 => x = 4
Thus, the critical points are x = -6 and x = 4. These points are essential because they divide the number line into intervals where the expression (x + 6)(x - 4) can change its sign. Identifying these critical points is a crucial step in solving quadratic inequalities.
Step 3: Test Intervals
The critical points divide the number line into three intervals: (-∞, -6), (-6, 4), and (4, ∞). We need to test a value from each interval in the inequality (x + 6)(x - 4) < 0 to determine where the inequality holds true.
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Interval (-∞, -6): Choose x = -7
((-7) + 6)((-7) - 4) = (-1)(-11) = 11 > 0
The inequality is not satisfied in this interval.
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Interval (-6, 4): Choose x = 0
(0 + 6)(0 - 4) = (6)(-4) = -24 < 0
The inequality is satisfied in this interval.
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Interval (4, ∞): Choose x = 5
(5 + 6)(5 - 4) = (11)(1) = 11 > 0
The inequality is not satisfied in this interval.
Step 4: Write the Solution
The interval (-6, 4) is where the inequality x² + 2x - 24 < 0 holds true. Therefore, the solution to the inequality is:
-6 < x < 4
In interval notation, the solution is (-6, 4). This means that any value of x between -6 and 4 (exclusive) will satisfy the original quadratic inequality. The process of testing intervals is a robust method to ensure you capture the correct solution set for any quadratic inequality.
Next, let's solve the second quadratic inequality: x² + 5x - 14 < 0. This process mirrors the one used previously, but with a different quadratic expression. The key steps remain the same: factor the quadratic, find the critical points, and test the intervals.
Step 1: Factor the Quadratic Expression
We need to factor the expression x² + 5x - 14. Look for two numbers that multiply to -14 and add to 5. These numbers are 7 and -2. Thus, we can factor the quadratic as:
x² + 5x - 14 = (x + 7)(x - 2)
Effective factoring is a cornerstone of solving quadratic inequalities. A solid grasp of factoring techniques will greatly enhance your ability to solve these types of problems efficiently.
Step 2: Find the Critical Points
Set each factor equal to zero and solve for x to find the critical points:
x + 7 = 0 => x = -7 x - 2 = 0 => x = 2
The critical points are x = -7 and x = 2. These critical points are where the expression (x + 7)(x - 2) changes its sign, so they are crucial for determining the intervals where the quadratic inequality holds.
Step 3: Test Intervals
The critical points divide the number line into three intervals: (-∞, -7), (-7, 2), and (2, ∞). We'll test a value from each interval in the inequality (x + 7)(x - 2) < 0.
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Interval (-∞, -7): Choose x = -8
((-8) + 7)((-8) - 2) = (-1)(-10) = 10 > 0
The inequality is not satisfied in this interval.
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Interval (-7, 2): Choose x = 0
(0 + 7)(0 - 2) = (7)(-2) = -14 < 0
The inequality is satisfied in this interval.
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Interval (2, ∞): Choose x = 3
(3 + 7)(3 - 2) = (10)(1) = 10 > 0
The inequality is not satisfied in this interval.
Step 4: Write the Solution
The interval (-7, 2) satisfies the inequality x² + 5x - 14 < 0. Therefore, the solution is:
-7 < x < 2
In interval notation, the solution is (-7, 2). This interval represents all x values that make the original quadratic inequality true. Through the systematic process of testing intervals, we confidently arrive at the correct solution set.
Now, let's address the third quadratic inequality: x² - 2x - 24 < 0. We will follow the same methodology as before: factor the quadratic expression, find the critical points, and test intervals to determine the solution.
Step 1: Factor the Quadratic Expression
Factor the quadratic x² - 2x - 24. We need two numbers that multiply to -24 and add to -2. These numbers are -6 and 4. So, the factored form is:
x² - 2x - 24 = (x - 6)(x + 4)
Factoring is a critical skill. The ability to quickly and accurately factor quadratic expressions is essential for efficiently solving quadratic inequalities.
Step 2: Find the Critical Points
Set each factor to zero to find the critical points:
x - 6 = 0 => x = 6 x + 4 = 0 => x = -4
The critical points are x = 6 and x = -4. These critical points mark the boundaries where the expression (x - 6)(x + 4) may change sign, which is key to solving the quadratic inequality.
Step 3: Test Intervals
The critical points divide the number line into three intervals: (-∞, -4), (-4, 6), and (6, ∞). Test a value from each interval in the inequality (x - 6)(x + 4) < 0.
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Interval (-∞, -4): Choose x = -5
((-5) - 6)((-5) + 4) = (-11)(-1) = 11 > 0
The inequality is not satisfied in this interval.
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Interval (-4, 6): Choose x = 0
(0 - 6)(0 + 4) = (-6)(4) = -24 < 0
The inequality is satisfied in this interval.
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Interval (6, ∞): Choose x = 7
(7 - 6)(7 + 4) = (1)(11) = 11 > 0
The inequality is not satisfied in this interval.
Step 4: Write the Solution
The interval (-4, 6) satisfies the inequality x² - 2x - 24 < 0. Thus, the solution is:
-4 < x < 6
In interval notation, the solution is (-4, 6). This represents the set of x values that make the original quadratic inequality true. The consistent application of testing intervals ensures an accurate solution.
Finally, let's solve the fourth quadratic inequality: x² - 9x + 20 < 0. We will continue to use the same method: factoring the quadratic, finding critical points, and testing intervals.
Step 1: Factor the Quadratic Expression
Factor the expression x² - 9x + 20. We need two numbers that multiply to 20 and add to -9. These numbers are -4 and -5. Therefore, the factored form is:
x² - 9x + 20 = (x - 4)(x - 5)
Mastering factoring techniques is crucial for solving quadratic inequalities efficiently. Being able to quickly identify the factors simplifies the subsequent steps.
Step 2: Find the Critical Points
Find the critical points by setting each factor equal to zero:
x - 4 = 0 => x = 4 x - 5 = 0 => x = 5
The critical points are x = 4 and x = 5. These critical points are where the expression (x - 4)(x - 5) changes its sign, and they are essential for determining the intervals where the quadratic inequality holds.
Step 3: Test Intervals
The critical points divide the number line into three intervals: (-∞, 4), (4, 5), and (5, ∞). Test a value from each interval in the inequality (x - 4)(x - 5) < 0.
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Interval (-∞, 4): Choose x = 3
(3 - 4)(3 - 5) = (-1)(-2) = 2 > 0
The inequality is not satisfied in this interval.
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Interval (4, 5): Choose x = 4.5
(4. 5 - 4)(4. 5 - 5) = (0. 5)(-0. 5) = -0. 25 < 0
The inequality is satisfied in this interval.
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Interval (5, ∞): Choose x = 6
(6 - 4)(6 - 5) = (2)(1) = 2 > 0
The inequality is not satisfied in this interval.
Step 4: Write the Solution
The interval (4, 5) satisfies the inequality x² - 9x + 20 < 0. Therefore, the solution is:
4 < x < 5
In interval notation, the solution is (4, 5). This represents the x values for which the quadratic inequality is true. The systematic method of testing intervals ensures we arrive at the correct solution.
In this comprehensive guide, we have explored the process of solving quadratic inequalities. We covered four specific examples, each demonstrating the essential steps:
- Factoring the quadratic expression.
- Finding the critical points by setting the factors to zero.
- Testing intervals on the number line to determine where the inequality holds.
- Writing the solution in interval notation.
By mastering these techniques, you can confidently tackle a wide range of quadratic inequalities. Understanding these concepts not only enhances your problem-solving skills but also provides a strong foundation for more advanced mathematical topics. Remember, practice is key. The more you work with these types of problems, the more proficient you will become. Keep practicing, and you'll master quadratic inequalities in no time!
