Solving Quadratic Inequalities Step-by-Step Guide

by Brainly ES FTUNILA 50 views
Iklan Headers

Hey guys! Ever struggled with quadratic inequalities? Don't worry, you're not alone! These types of problems can seem tricky at first, but with the right approach, they become super manageable. In this article, we're going to break down how to solve quadratic inequalities step-by-step, showing you how to express your answers in set notation, interval notation, and graphically. So, grab your pencils and let's dive in!

What are Quadratic Inequalities?

Before we jump into solving, let’s make sure we're all on the same page. A quadratic inequality is an inequality that involves a quadratic expression. Remember, a quadratic expression is one of the form ax² + bx + c, where a, b, and c are constants and a ≠ 0. An inequality, on the other hand, is a mathematical statement that compares two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). So, putting it all together, a quadratic inequality looks something like ax² + bx + c < 0, ax² + bx + c > 0, ax² + bx + c ≤ 0, or ax² + bx + c ≥ 0. Solving these inequalities means finding the set of all x values that make the inequality true. This is crucial in various fields, such as physics, engineering, and economics, where understanding the range of possible solutions is more practical than finding a single answer. For example, in physics, we might want to find the range of initial velocities that would allow a projectile to reach a certain height, or in economics, we might be interested in the range of prices that would result in a profit. Understanding the underlying concepts and techniques for solving quadratic inequalities provides a solid foundation for tackling more complex problems in these areas.

Why are Quadratic Inequalities Important?

You might be thinking, "Okay, that's the definition, but why should I care?" Well, quadratic inequalities pop up in all sorts of real-world situations! They're used in physics to model projectile motion, in engineering to design structures, and even in economics to analyze profit margins. Understanding how to solve them gives you a powerful tool for problem-solving in many different fields. In real-world applications, quadratic inequalities are incredibly versatile. For instance, in physics, they can help determine the range of initial velocities needed for a projectile to reach a specific height. Imagine you're launching a rocket; you'd need to know the range of velocities that would allow it to reach its intended destination without falling short or overshooting. Similarly, in engineering, quadratic inequalities can be used to design structures that can withstand certain stresses or loads. For example, when building a bridge, engineers need to ensure that the structure can handle a range of weights and forces without collapsing. In economics, quadratic inequalities can be applied to analyze profit margins. A business owner might use them to determine the range of prices that would result in a profit, taking into account factors like production costs and demand. In essence, mastering quadratic inequalities opens up a world of possibilities in various disciplines, providing a valuable skill set for problem-solving in both academic and professional contexts. By understanding how these inequalities work, you can tackle a wide array of real-world challenges with greater confidence and accuracy. This practical relevance is what makes learning about quadratic inequalities not just an academic exercise, but a valuable tool for critical thinking and problem-solving.

Solving Quadratic Inequalities: A Step-by-Step Guide

Alright, let's get down to business! Here’s a general method for solving quadratic inequalities:

  1. Rewrite the Inequality: First, rewrite the inequality so that one side is zero. This means you'll want to get it into the form ax² + bx + c < 0, ax² + bx + c > 0, ax² + bx + c ≤ 0, or ax² + bx + c ≥ 0. This step is crucial because it sets the stage for finding the critical points of the quadratic expression, which are the points where the expression equals zero. By rearranging the inequality to have zero on one side, we make it easier to identify these critical points. For instance, if you have the inequality x² - 3x > 4, the first step is to subtract 4 from both sides to get x² - 3x - 4 > 0. This form allows us to focus on the behavior of the quadratic expression x² - 3x - 4 relative to zero. Understanding when the expression is positive, negative, or zero helps us determine the intervals that satisfy the inequality. This initial rearrangement is a fundamental step in the process, laying the groundwork for the subsequent steps and making the inequality more manageable to solve. It ensures that we are comparing the quadratic expression to a clear benchmark, which is zero, thereby simplifying the analysis and leading to an accurate solution.

  2. Find the Roots (Critical Points): Next, find the roots of the corresponding quadratic equation ax² + bx + c = 0. You can do this by factoring, using the quadratic formula, or completing the square. The roots, also known as the critical points, are the values of x where the quadratic expression equals zero. These points are critical because they divide the number line into intervals, within each of which the quadratic expression has a consistent sign (either positive or negative). This is because the quadratic expression can only change its sign at its roots. Factoring is often the quickest method if the quadratic expression is easily factorable. For example, if we have x² - 5x + 6 = 0, we can factor it into (x - 2)(x - 3) = 0, giving us the roots x = 2 and x = 3. If factoring isn't straightforward, the quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), is a reliable alternative. Completing the square is another method, particularly useful for understanding the vertex form of the quadratic. Regardless of the method used, finding the roots is a crucial step. These roots serve as boundaries on the number line, helping us identify the intervals where the quadratic expression is either positive or negative, which is essential for solving the inequality. The accuracy of these roots directly impacts the correct solution set for the inequality.

  3. Create a Number Line and Test Intervals: Draw a number line and mark the roots you found in the previous step. These roots divide the number line into intervals. Choose a test value from each interval and plug it into the original inequality. If the test value satisfies the inequality, then that entire interval is part of the solution. This step is crucial for determining the regions on the number line where the inequality holds true. By plotting the roots (critical points) on the number line, we effectively partition the line into distinct intervals. Within each interval, the quadratic expression will maintain a consistent sign—either always positive or always negative—because the expression can only change signs at its roots. To ascertain the sign within each interval, we select a test value—any number within that interval—and substitute it into the original inequality. For instance, if our roots are x = 2 and x = 3, we would have three intervals to test: (-∞, 2), (2, 3), and (3, ∞). We might choose 0 as a test value for the first interval, 2.5 for the second, and 4 for the third. By plugging these values into the original inequality, we can determine whether the inequality is satisfied in each interval. If the test value satisfies the inequality, it means that all values in that interval will also satisfy it, making that interval part of the solution set. Conversely, if the test value does not satisfy the inequality, the interval is excluded from the solution set. This process ensures that we accurately identify all regions on the number line that meet the conditions of the inequality, providing a comprehensive solution.

  4. Write the Solution in Different Notations: Finally, write the solution using set notation, interval notation, and graphically. Set notation is a way of writing down sets of numbers using curly braces and symbols like union (∪) and intersection (∩). Interval notation uses parentheses and brackets to indicate intervals on the number line. Parentheses mean the endpoint is not included, while brackets mean it is. Graphically, you represent the solution by shading the intervals on the number line that satisfy the inequality. This final step is essential for expressing the solution in a clear and understandable manner across various mathematical contexts. Set notation provides a formal way to describe the solution set, using curly braces to enclose the elements or intervals. For example, the set of all x such that 2 < x ≤ 5 can be written as {x | 2 < x ≤ 5}. This notation is particularly useful when dealing with complex solutions that may involve multiple intervals or specific points. Interval notation, on the other hand, offers a concise way to represent intervals on the number line. It uses parentheses and brackets to denote whether the endpoints are included or excluded from the solution. A parenthesis indicates that the endpoint is not included (i.e., an open interval), while a bracket indicates that the endpoint is included (i.e., a closed interval). For instance, the interval (2, 5] represents all numbers greater than 2 and less than or equal to 5. When solutions involve multiple intervals, the union symbol (∪) is used to combine them. Graphically, the solution is represented by shading the intervals on the number line that satisfy the inequality. Open circles are used to mark endpoints that are not included (corresponding to parentheses), and closed circles or brackets are used to mark endpoints that are included (corresponding to brackets). This visual representation offers an intuitive understanding of the solution set, making it easy to see the range of values that satisfy the inequality. Together, these three notations provide a comprehensive way to express and interpret the solution to a quadratic inequality.

Let's Solve Some Inequalities!

Now, let's put this method into practice by solving the inequalities you provided:

1. x² - 12 < x

  • Step 1: Rewrite the Inequality: Subtract x from both sides to get x² - x - 12 < 0.

  • Step 2: Find the Roots: Factor the quadratic expression: (x - 4)(x + 3) = 0. The roots are x = 4 and x = -3.

  • Step 3: Create a Number Line and Test Intervals: Draw a number line with -3 and 4 marked. We have three intervals to test: (-∞, -3), (-3, 4), and (4, ∞).

    • Test x = -4 (in (-∞, -3)): (-4)² - (-4) - 12 = 16 + 4 - 12 = 8, which is not less than 0.
    • Test x = 0 (in (-3, 4)): (0)² - (0) - 12 = -12, which is less than 0.
    • Test x = 5 (in (4, ∞)): (5)² - (5) - 12 = 25 - 5 - 12 = 8, which is not less than 0.

    So, the interval (-3, 4) satisfies the inequality.

  • Step 4: Write the Solution in Different Notations:

    • Set Notation: {x | -3 < x < 4}
    • Interval Notation: (-3, 4)
    • Graphical Notation: A number line shaded between -3 and 4, with open circles at -3 and 4.

2. 3x² + 10x >= 8

  • Step 1: Rewrite the Inequality: Subtract 8 from both sides to get 3x² + 10x - 8 ≥ 0.

  • Step 2: Find the Roots: Factor the quadratic expression: (3x - 2)(x + 4) = 0. The roots are x = 2/3 and x = -4.

  • Step 3: Create a Number Line and Test Intervals: Draw a number line with -4 and 2/3 marked. We have three intervals to test: (-∞, -4), (-4, 2/3), and (2/3, ∞).

    • Test x = -5 (in (-∞, -4)): 3(-5)² + 10(-5) - 8 = 75 - 50 - 8 = 17, which is greater than or equal to 0.
    • Test x = 0 (in (-4, 2/3)): 3(0)² + 10(0) - 8 = -8, which is not greater than or equal to 0.
    • Test x = 1 (in (2/3, ∞)): 3(1)² + 10(1) - 8 = 3 + 10 - 8 = 5, which is greater than or equal to 0.

    So, the intervals (-∞, -4] and [2/3, ∞) satisfy the inequality.

  • Step 4: Write the Solution in Different Notations:

    • Set Notation: {x | x ≤ -4} ∪ {x | x ≥ 2/3}
    • Interval Notation: (-∞, -4] ∪ [2/3, ∞)
    • Graphical Notation: A number line shaded to the left of -4 (including -4) and to the right of 2/3 (including 2/3).

3. x² > 7x - 10

  • Step 1: Rewrite the Inequality: Subtract 7x and add 10 to both sides to get x² - 7x + 10 > 0.

  • Step 2: Find the Roots: Factor the quadratic expression: (x - 2)(x - 5) = 0. The roots are x = 2 and x = 5.

  • Step 3: Create a Number Line and Test Intervals: Draw a number line with 2 and 5 marked. We have three intervals to test: (-∞, 2), (2, 5), and (5, ∞).

    • Test x = 0 (in (-∞, 2)): (0)² - 7(0) + 10 = 10, which is greater than 0.
    • Test x = 3 (in (2, 5)): (3)² - 7(3) + 10 = 9 - 21 + 10 = -2, which is not greater than 0.
    • Test x = 6 (in (5, ∞)): (6)² - 7(6) + 10 = 36 - 42 + 10 = 4, which is greater than 0.

    So, the intervals (-∞, 2) and (5, ∞) satisfy the inequality.

  • Step 4: Write the Solution in Different Notations:

    • Set Notation: {x | x < 2} ∪ {x | x > 5}
    • Interval Notation: (-∞, 2) ∪ (5, ∞)
    • Graphical Notation: A number line shaded to the left of 2 (excluding 2) and to the right of 5 (excluding 5).

4. 4x² + 9x - 9 < 0

  • Step 1: Rewrite the Inequality: The inequality is already in the correct form: 4x² + 9x - 9 < 0.

  • Step 2: Find the Roots: Factor the quadratic expression: (4x - 3)(x + 3) = 0. The roots are x = 3/4 and x = -3.

  • Step 3: Create a Number Line and Test Intervals: Draw a number line with -3 and 3/4 marked. We have three intervals to test: (-∞, -3), (-3, 3/4), and (3/4, ∞).

    • Test x = -4 (in (-∞, -3)): 4(-4)² + 9(-4) - 9 = 64 - 36 - 9 = 19, which is not less than 0.
    • Test x = 0 (in (-3, 3/4)): 4(0)² + 9(0) - 9 = -9, which is less than 0.
    • Test x = 1 (in (3/4, ∞)): 4(1)² + 9(1) - 9 = 4 + 9 - 9 = 4, which is not less than 0.

    So, the interval (-3, 3/4) satisfies the inequality.

  • Step 4: Write the Solution in Different Notations:

    • Set Notation: {x | -3 < x < 3/4}
    • Interval Notation: (-3, 3/4)
    • Graphical Notation: A number line shaded between -3 and 3/4, with open circles at -3 and 3/4.

5. (x - 4)(x + 1) <= 0

  • Step 1: Rewrite the Inequality: The inequality is already in factored form and has zero on one side.

  • Step 2: Find the Roots: The roots are x = 4 and x = -1.

  • Step 3: Create a Number Line and Test Intervals: Draw a number line with -1 and 4 marked. We have three intervals to test: (-∞, -1), (-1, 4), and (4, ∞).

    • Test x = -2 (in (-∞, -1)): (-2 - 4)(-2 + 1) = (-6)(-1) = 6, which is not less than or equal to 0.
    • Test x = 0 (in (-1, 4)): (0 - 4)(0 + 1) = (-4)(1) = -4, which is less than or equal to 0.
    • Test x = 5 (in (4, ∞)): (5 - 4)(5 + 1) = (1)(6) = 6, which is not less than or equal to 0.

    So, the interval [-1, 4] satisfies the inequality.

  • Step 4: Write the Solution in Different Notations:

    • Set Notation: {x | -1 ≤ x ≤ 4}
    • Interval Notation: [-1, 4]
    • Graphical Notation: A number line shaded between -1 and 4, with closed circles at -1 and 4.

6. x² >= x + 6

  • Step 1: Rewrite the Inequality: Subtract x and 6 from both sides to get x² - x - 6 ≥ 0.

  • Step 2: Find the Roots: Factor the quadratic expression: (x - 3)(x + 2) = 0. The roots are x = 3 and x = -2.

  • Step 3: Create a Number Line and Test Intervals: Draw a number line with -2 and 3 marked. We have three intervals to test: (-∞, -2), (-2, 3), and (3, ∞).

    • Test x = -3 (in (-∞, -2)): (-3)² - (-3) - 6 = 9 + 3 - 6 = 6, which is greater than or equal to 0.
    • Test x = 0 (in (-2, 3)): (0)² - (0) - 6 = -6, which is not greater than or equal to 0.
    • Test x = 4 (in (3, ∞)): (4)² - (4) - 6 = 16 - 4 - 6 = 6, which is greater than or equal to 0.

    So, the intervals (-∞, -2] and [3, ∞) satisfy the inequality.

  • Step 4: Write the Solution in Different Notations:

    • Set Notation: {x | x ≤ -2} ∪ {x | x ≥ 3}
    • Interval Notation: (-∞, -2] ∪ [3, ∞)
    • Graphical Notation: A number line shaded to the left of -2 (including -2) and to the right of 3 (including 3).

7. x² <= x

  • Step 1: Rewrite the Inequality: Subtract x from both sides to get x² - x ≤ 0.

  • Step 2: Find the Roots: Factor the quadratic expression: x(x - 1) = 0. The roots are x = 0 and x = 1.

  • Step 3: Create a Number Line and Test Intervals: Draw a number line with 0 and 1 marked. We have three intervals to test: (-∞, 0), (0, 1), and (1, ∞).

    • Test x = -1 (in (-∞, 0)): (-1)² - (-1) = 1 + 1 = 2, which is not less than or equal to 0.
    • Test x = 0.5 (in (0, 1)): (0.5)² - (0.5) = 0.25 - 0.5 = -0.25, which is less than or equal to 0.
    • Test x = 2 (in (1, ∞)): (2)² - (2) = 4 - 2 = 2, which is not less than or equal to 0.

    So, the interval [0, 1] satisfies the inequality.

  • Step 4: Write the Solution in Different Notations:

    • Set Notation: {x | 0 ≤ x ≤ 1}
    • Interval Notation: [0, 1]
    • Graphical Notation: A number line shaded between 0 and 1, with closed circles at 0 and 1.

Conclusion

There you have it! Solving quadratic inequalities can seem daunting, but by following these steps, you can tackle them like a pro. Remember to rewrite the inequality, find the roots, create a number line, test intervals, and write your solution in different notations. Keep practicing, and you'll become a quadratic inequality master in no time!

If you guys have any questions or want to try more examples, let me know in the comments below. Happy solving!