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Lesson Plan: Quadratic Inequalities

 

Lesson Summary

This 50-minute lesson focuses on solving quadratic inequalities both algebraically and graphically. Students will learn to interpret and represent solutions using interval notation. The lesson incorporates multimedia resources from Media4Math.com to enhance understanding of key concepts. Through warm-up activities, detailed instruction, and practical examples, students will develop skills in solving quadratic inequalities. The lesson concludes with a comprehensive review and a 10-question quiz with an accompanying answer key to assess student learning.

Lesson Objectives

  • Solve quadratic inequalities using algebraic methods
  • Solve quadratic inequalities using graphical methods
  • Represent solutions of quadratic inequalities using interval notation
  • Apply quadratic inequality concepts to real-world situations

Common Core Standards

  • HSA.REI.4b Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
  • HSA.REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Prerequisite Skills

  • Solving linear inequalities
  • Graphing quadratic functions
  • Identifying roots of quadratic equations

Key Vocabulary

 

 

Warm Up Activities

Choose from one or more of these activities.

Activity 1: Review Inequalities

Review inequalities using the following resources: 

Activity 2: View Review of Inequalities

Watch this video on linear inequalities: 

Activity 3: Using Desmos

Graph a linear inequality on Desmos.com. Follow these steps:

  1. Open Desmos.com
  2. Enter a linear inequality (e.g., y > 2x + 1)
  3. Observe the shaded region representing the solution
  4. Experiment with different inequalities

 

 

Teach (25 minutes)

Introduction to Quadratic Inequalities

In this section, we will explore quadratic inequalities, an extension of quadratic equations. A quadratic inequality involves a quadratic expression compared to a value or another expression using inequality symbols such as <, ≤, >, or ≥. These inequalities can be categorized into two types: single-variable quadratic inequalities and two-variable quadratic inequalities.

  • Single-variable quadratic inequalities: For example, x² - 4x + 3 > 0, where the solution is a range or ranges of x-values that satisfy the inequality. These solutions can be represented on a number line or in interval notation.
  • Two-variable quadratic inequalities: For example, y > x² - 4x + 3, where the solution is a region in the coordinate plane that satisfies the inequality. These solutions are often represented as shaded areas showing all the points (x, y) where the inequality holds true.

Before diving into quadratic inequalities, let’s briefly review inequalities in general. Inequalities express relationships where quantities are not equal, representing conditions like "greater than" or "less than." Solutions to inequalities can be expressed in multiple formats—using a number line for single-variable inequalities or graphically for two-variable ones.

But why study quadratic inequalities? These concepts help us model and solve real-world problems involving constraints or limits, such as determining feasible regions in optimization problems, analyzing projectile motion, or understanding economic models. Studying quadratic inequalities enhances our reasoning skills by enabling us to interpret and work with ranges of values rather than specific solutions, making them an essential tool in algebra and beyond.

This section introduces quadratic inequalities, focusing on both single-variable and two-variable examples. Students will learn to solve these inequalities using algebraic and graphical methods, applying their knowledge to real-world scenarios. The instruction emphasizes the importance of understanding the relationship between the inequality sign and the parabola's shape when determining solution regions.

Example 1

Solve x² - 4x - 5 > 0

Step 1: Factor the quadratic expression 

x² - 4x - 5 = (x - 5)(x + 1)

Step 2: Find the roots 

x = 5 or x = -1

Step 3: Create a sign chart 

(-∞, -1): (+)(-) Negative 
(-1, 5): (-)(-) Positive 
(5, ∞): (+)(+) Positive

Step 4: Identify the intervals where the inequality is true

x < -1 or x > 5

Step 5: Write the solution in interval notation 

(-∞, -1) ∪ (5, ∞)

 

Quadratics

 

Compare this to the graph showing the parabola:

 

Quadratics

 

Summary: This example demonstrates how to solve a quadratic inequality by factoring, finding roots, and using a sign chart to determine the solution intervals.

Example 2

Solve -2x² + 12x - 10 ≤ 0

Step 1: Factor the quadratic expression 

-2(x² - 6x + 5) = -2(x - 1)(x - 5)

Step 2: Find the roots 

x = 1 or x = 5

Step 3: Consider the inequality sign and the negative coefficient of x² 


The parabola opens downward, so the solution is between the roots

Step 4: Write the solution in interval notation 

[1, 5]

Quadratics

 

Summary: This example illustrates how the direction of the inequality and the parabola's orientation affect the solution, resulting in a closed interval between the roots.

Example 3 (Real-world application)

A company's profit P (in thousands of dollars) for producing and selling x units of a product is given by P(x) = -0.01x² + 40x - 30000. For what production levels will the company make a profit?

Step 1: Set up the inequality 

-0.01x² + 40x - 30000 > 0

Step 2: Solve the quadratic equation 

-0.01x² + 40x - 30000 = 0 
x = 1000 or x = 3000

Step 3: Analyze the parabola 

The parabola opens downward, so the profit is positive between the roots

Step 4: Write the solution in interval notation 

(1000, 3000)

Step 5: Interpret the result 

The company will make a profit when producing between 1000 and 3000 units.

 

Quadratics

 

Summary: This real-world example shows how quadratic inequalities can be applied to business scenarios, determining the range of production levels that result in a profit.

Example 4: Designing a Water Fountain

 

Quadratics

 

An urban park designer is working on a new water fountain feature where water is sprayed upward in a parabolic trajectory. The trajectory of the water can be modeled by the equation:

y = -½x² + 4x

where x and y are measured in feet, with x being the horizontal distance from the fountain's base and y being the height of the water stream.

The designer needs to ensure that the water stays within a certain boundary to avoid overspraying into walkways. The maximum height the water can reach should not exceed 6 feet, and the horizontal spray distance should not exceed 10 feet. This scenario can be modeled by the quadratic inequality:

y ≤ -½x² + 4x, where 0 ≤ x ≤ 10 and y ≤ 6.

Task:

Determine the region in the coordinate plane where the fountain’s water trajectory satisfies these constraints.

Solution Steps:

  1. Graph the Quadratic Equation: Plot the parabola y = -½x² + 4x. The vertex occurs at x = -b/2a = 4, with the maximum height of the water at (4, 8). However, since y ≤ 6, the actual trajectory is limited to heights at or below 6 feet.

  2. Apply the Height Constraint: To incorporate the y ≤ 6 constraint, draw a horizontal line at y = 6. The intersection points of y = 6 with y = -½x² + 4x are found by solving:

    6 = -½x² + 4x

    Simplify to:

    ½x² - 4x + 6 = 0

    Multiply through by 2:

    x² - 8x + 12 = 0

    Factor:

    (x - 6)(x - 2) = 0

    So, x = 2 and x = 6. The trajectory remains valid between these points, 2 ≤ x ≤ 6.

  3. Apply the Horizontal Distance Constraint: The domain of the trajectory is limited to 0 ≤ x ≤ 10. Combine this with the solution for x from step 2 to get:

    2 ≤ x ≤ 6.

  4. Combine Constraints: The region satisfying all conditions is the part of the parabola between 2 ≤ x ≤ 6 and below y = 6. Shade this region on the graph to represent the solution.

 

Quadratics

 

Discussion

This example demonstrates how quadratic inequalities in two variables can model practical situations, such as limiting trajectories for safety or efficiency in design.

 

Review (10 minutes)

In this lesson, we explored quadratic inequalities, learning to solve them both algebraically and graphically. We practiced representing solutions using interval notation, which is crucial for clear communication of mathematical results. The lesson addressed our objectives by demonstrating various solution methods and applying them to real-world scenarios.

Key vocabulary review

  • Inequality: Relates quantities that are not equal
  • Solution set: All values satisfying the inequality
  • Interval notation: Concise way to represent solution sets
  • Quadratic equation: Basis for quadratic inequalities
  • Roots: Critical points in solving quadratic inequalities

Additional examples:

Example 1

Graphical solution: y > x² - 4x + 3

Graph y = x² - 4x + 3 and shade the region above the parabola. The solution is the shaded area above the parabola.

Summary: This example demonstrates how to solve a two-variable quadratic inequality graphically, emphasizing the importance of visualizing the solution region.

 

Quadratics

 

Example 2 (Real-world application)

A ball is thrown upward from a 5-meter high platform with an initial velocity of 15 m/s. The height h (in meters) after t seconds is given by h = -4.9t² + 15t + 5. For how long will the ball be above 10 meters high?

Set up the inequality: -4.9t² + 15t + 5 > 10 
Solve: -4.9t² + 15t - 5 > 0 
Factoring: -(4.9t² - 15t + 5) > 0 
Roots: t ≈ 0.38 or t ≈ 2.69 
Solution: (0.37, 2.69) seconds

Quadratics

The ball will be above 10 meters high for approximately 2.32 seconds.

Summary: This real-world example applies quadratic inequalities to physics, determining the time interval during which an object remains above a certain height.

 

 

Quiz

Directions: Solve the following quadratic inequalities. Show your work and express your answers using interval notation where appropriate.

  1. x² + 3x - 10 < 0

     
  2. 2x² - 5x - 3 ≥ 0

     
  3. -x² + 6x - 5 > 0

     
  4. x² + 2x - 8 ≤ 0

     
  5. 3x² - 12x + 9 < 0

     
  6. A rectangle has a perimeter of 24 units. For what range of lengths will the area of the rectangle be greater than 32 square units?

     
  7. Solve the inequality graphically: y < -x² + 4x - 3

     
  8. The height of a projectile (in meters) t seconds after it is launched is given by h(t) = -5t² + 40t. For how long will the projectile be above 75 meters?

     
  9. Solve: |x² - 4| > 5

     
  10. For what values of x is the parabola y = x² - 6x + 5 below the x-axis?

Answer Key

  1. (-5, 2)
  2. (-∞, -1] ∪ [1.5, ∞)
  3. (1, 5)
  4. [-4, 2]
  5. (1, 3)
  6. (4, 8) units
  7. The region below the parabola y = -x² + 4x - 3
  8. (2, 6) seconds
  9. (-∞, -3) ∪ (-√9, √9) ∪ (3, ∞)
  10. (2, 4)