Lesson 2.4: Quadratic Inequalities and Modelling with Equations
Introduction
In this lesson, we will dive into the topic of quadratic inequalities and how they can be solved using sketches and sign analysis. Additionally, we will cover how to represent combined solution sets and use equations and inequalities to model real-world situations in business and science contexts. By the end of this lesson, students, you will understand how to confidently tackle quadratic inequalities, create sketches, and express solutions as intervals.
Learning Objectives
- Solve quadratic inequalities using sketches and sign analysis.
- Represent combined solution sets correctly.
- Use equations and inequalities to model simple business and science contexts.
- Solve a quadratic inequality and express the solution as an interval.
- Represent a combined solution set on a number line.
Quadratic Inequalities
A quadratic inequality is an inequality that involves a quadratic expression. The general form is given by:
$$ ax^2 + bx + c < 0 $$
$$ ax^2 + bx + c > 0 $$
$$ ax^2 + bx + c \leq 0 $$
$$ ax^2 + bx + c \geq 0 $$
where $ a, b, c $ are real numbers, and a
eq 0 . To solve a quadratic inequality, you can follow these steps:
Step 1: Solve the Corresponding Quadratic Equation
The first step in solving a quadratic inequality is to determine where the quadratic expression equals zero by solving the corresponding quadratic equation:
$$ ax^2 + bx + c = 0 $$
You can use the quadratic formula:
$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$
Example 1:
Let's solve the quadratic inequality:
$$ 2x^2 - 4x - 6 < 0 $$
First, we solve the equation:
$$ 2x^2 - 4x - 6 = 0 $$
Using the quadratic formula:
$$ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(2)(-6)}}{2(2)} $$
Calculate the discriminant:
$$ b^2 - 4ac = 16 + 48 = 64 $$
Substituting back into the formula:
$$ x = \frac{4 \pm \sqrt{64}}{4} = \frac{4 \pm 8}{4} $$
Thus, we get:
$$ x_1 = 3 \quad \text{and} \quad x_2 = -1 $$
Step 2: Determine Test Intervals
Next, identify the intervals defined by the roots $ x_1 $ and $ x_2 $. The critical points split the number line into intervals that we will test:
- $ (-\infty, -1) $
- $ (-1, 3) $
- $ (3, \infty) $
Step 3: Choose Test Points
Select a test point from each interval to determine if the inequality holds in that interval.
- For $ (-\infty, -1) $, let’s use $ x = -2 $:
$$ 2(-2)^2 - 4(-2) - 6 = 8 + 8 - 6 = 10 > 0 $$
- For $ (-1, 3) $, let’s use $ x = 0 $:
$$ 2(0)^2 - 4(0) - 6 = -6 < 0 $$
- For $ (3, \infty) $, let’s use $ x = 4 $:
$$ 2(4)^2 - 4(4) - 6 = 32 - 16 - 6 = 10 > 0 $$
Step 4: Assemble the Solution
From the tests, we find that the quadratic expression is negative between $ -1 $ and $ 3 $. Therefore, the solution to the inequality $ 2x^2 - 4x - 6 < 0 $ is:
$$ x \in (-1, 3) $$
Graphical Representation
To enhance understanding, sketch the graph of the quadratic function:
- Identify the vertex and direction (opening up or down based on the sign of $ a $).
- Plot the points where the quadratic equals zero (the roots) and illustrate the regions where the inequality holds.
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Working with Combined Inequalities
When dealing with combined inequalities, we often need to find overlap between different inequalities. The key is to understand how to represent each individual solution and then determine their intersection.
Example 2:
Suppose we have two inequalities:
- $ x^2 - 4 < 0 $
- $ x - 1 \geq 0 $
Step 1: Solve $ x^2 - 4 < 0 $
This can be rearranged as:
$$ (x - 2)(x + 2) < 0 $$
The roots are $ x = -2 $ and $ x = 2 $. Testing intervals gives:
- $ (-\infty, -2) $: Positive
- $ (-2, 2) $: Negative
- $ (2, \infty) $: Positive
Thus, the solution is:
$$ x \in (-2, 2) $$
Step 2: Solve $ x - 1 \geq 0 $
This means:
$$ x \geq 1 $$
Step 3: Combine Solutions
To find the combined solution set, we need to see the intersection:
- From the first inequality, we know $ x \in (-2, 2) $
- From the second inequality, we know $ x \in [1, \infty) $
The intersection is:
$$ x \in [1, 2) $$
Final Graphical Representation
On a number line, represent both intervals, highlighting the intersection where they meet, and ensure to label them clearly.
Modelling with Quadratic Equations and Inequalities
Quadratic equations and inequalities can also be used to model real-world scenarios, particularly in business and science contexts. Quadratic models can help analyze situations like maximizing profit, determining projectile motion, and more.
Example 3: Business Context
Assume a company finds that their profit $ P(x) $ from selling $ x $ items can be modeled by:
$$ P(x) = -5x^2 + 150x - 200 $$
To find how many items to sell to maximize profit, we can complete the square or use the vertex formula. For a quadratic in the form:
$$ ax^2 + bx + c $$
The vertex occurs at:
$$ x = -\frac{b}{2a} $$
In this case:
$$ x = -\frac{150}{2 \cdot -5} = 15 $$
The company maximizes profit when they sell 15 items. To model conditions for profit greater than a certain value, set inequalities based on profit calculations.
Example 4: Science Context
In physics, a projectile's height $ h(t) $ can be given by:
$$ h(t) = -4.9t^2 + v_0 t + h_0 $$
where $ v_0 $ is the initial velocity and $ h_0 $ is the initial height. Using these equations allows us to analyze the projectile's motion, determining when it is above a certain height or when it hits the ground.
Conclusion
In this lesson, students, we explored the solving of quadratic inequalities, the importance of sketches, and the representation of solution sets. We also learned how to model real-world scenarios using quadratic equations and inequalities. With these concepts, you are better equipped to apply mathematics within various contexts.
Study Notes
- Quadratic inequalities take the form $ ax^2 + bx + c < 0 $ or similar.
- Solve for roots by setting the quadratic equation to zero: $ ax^2 + bx + c = 0 $.
- Use test points in intervals determined by critical points to find where the inequality holds.
- Combined solution sets involve finding intersections of individual solutions.
- Quadratic models can be applied in business and physical contexts to optimize outcomes.