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Interactive Audio Lesson
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Solving Quadratic Inequalities
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Today, we'll work on solving quadratic inequalities together through practice exercises. Can anyone remind us of the standard form for a quadratic inequality?
Isn’t it like 𝑎𝑥² + 𝑏𝑥 + 𝑐 < 0?
Exactly! Remember, we often write it as 𝑎𝑥² + 𝑏𝑥 + 𝑐 <, ≤, >, or ≥ 0. Now let's move on to the first exercise: Solve 𝑥² + 2𝑥 − 15 > 0.
I would factor it as (𝑥 + 5)(𝑥 - 3) > 0.
And the roots are -5 and 3, right?
Correct! Now, let’s discuss the intervals. Can anyone tell me how to analyze those intervals?
We choose test points in those intervals to see where the inequality holds!
Great! This helps us identify the valid solutions.
To recap, we factored the inequality, found roots, and will use test points now. Ready for the next one?
Using Graphs to Solve Inequalities
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Let’s now visualize the inequalities on a graph. Why is it helpful, do you think?
It helps us see if that parabola opens up or down and where it crosses the x-axis!
Exactly! The shape tells us a lot. Like in the second exercise, 𝑥² − 4𝑥 + 4 ≤ 0. Can someone solve it graphically?
I’d graph it, and since it's a perfect square, it touches the x-axis only at one point.
Right! So the solution can be written as just that one point.
That helps clear it up—since it only touches the x-axis once, it makes sense!
Excellent conversation! Let's summarize: support your algebra skills with graphical insights. Ready to try a word problem next?
Real-World Applications
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Quadratic inequalities aren't just about numbers. They apply to real-life problems! Who remembers a context where we use these?
Like projectile motion, right? To find when something is above a height?
Exactly! In our example, we had a ball with height modeled by ℎ(𝑡) = −5𝑡² + 20𝑡. Can anyone tell me how to find out when the ball is at least 15 meters high?
So we set up -5𝑡² + 20𝑡 ≥ 15?
Correct! By rearranging, it becomes a quadratic inequality we can solve. What can we say about our solutions?
We find times when it's true, such as when the ball rises and falls.
Excellent! Now we see how math interacts with the world. Let's wrap that up: whether it's physics or economics, quadratic inequalities let us model powerful scenarios.
Introduction & Overview
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Quick Overview
Standard
The section includes several exercises focused on solving quadratic inequalities, using various methods such as factorization and graphical representation. Solutions encourage students to engage with real-world applications.
Detailed
Practice Exercises
In this section, students will apply their understanding of quadratic inequalities through a series of practice exercises designed to develop their skills in solving and representing these inequalities. Focus will be placed on the various forms of quadratic inequalities, the necessary steps to solve them, and connecting these concepts to real-life scenarios. The exercises encourage critical thinking and reinforce the theoretical knowledge gained throughout the chapter.
Audio Book
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Practice Problem 1
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- Solve 𝑥² + 2𝑥 - 15 > 0
Detailed Explanation
To solve the inequality 𝑥² + 2𝑥 - 15 > 0, we first move all terms to one side and write the corresponding equation: 𝑥² + 2𝑥 - 15 = 0. Next, we factor the expression, which gives us (𝑥 + 5)(𝑥 - 3) = 0. The roots of this equation are 𝑥 = -5 and 𝑥 = 3. These roots divide the number line into intervals: (-∞, -5), (-5, 3), and (3, ∞). We will now test each interval using a test point to determine where the inequality holds true. For (-∞, -5), if we choose 𝑥 = -6, we get a positive value. For (-5, 3), if we choose 𝑥 = 0, we get a negative value. For (3, ∞), if we choose 𝑥 = 4, we get a positive value. Thus, the solution to the inequality is 𝑥 < -5 or 𝑥 > 3.
Examples & Analogies
Imagine you're measuring the temperature in a room, and you know that too cold (say below -5 degrees) and too hot (above 3 degrees) are uncomfortable. You can say, 'I want the temperature either below -5 or above 3' so that the room stays comfortable.
Practice Problem 2
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- Solve 𝑥² - 4𝑥 + 4 ≤ 0
Detailed Explanation
The inequality 𝑥² - 4𝑥 + 4 ≤ 0 can be approached by first rewriting it as an equation: 𝑥² - 4𝑥 + 4 = 0. We can factor this as (𝑥 - 2)(𝑥 - 2) = 0, which gives us a double root, 𝑥 = 2. This means that the quadratic touches the x-axis at just this one point and does not cross it. This divides the number line into intervals, but since we only have one root, we will analyze around 𝑥 = 2. Selecting test points, when 𝑥 < 2 (let's say 𝑥 = 1), we see that the expression is positive. When 𝑥 > 2 (for instance, 𝑥 = 3), the expression is also positive. Thus, the only solution is the point where the quadratic equals 0. Therefore, the answer is 𝑥 = 2.
Examples & Analogies
Think of this situation like a balance point. Imagine you're balancing on a seesaw, and the middle point is when you are perfectly still. The only time you're not going up or down (the 'zero gravity' point) is at 2. In all other positions, you're either tipping down (positive) or coming back up (not satisfying the inequality).
Practice Problem 3
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- Solve -𝑥² + 6𝑥 - 9 < 0
Detailed Explanation
To solve -𝑥² + 6𝑥 - 9 < 0, we begin with the equation -𝑥² + 6𝑥 - 9 = 0. This can be multiplied by -1 to simplify our calculations: 𝑥² - 6𝑥 + 9 = 0, which factors to (𝑥 - 3)(𝑥 - 3) = 0. Thus, we obtain a double root at 𝑥 = 3. The inequality states that we are interested in where the quadratic is less than 0. Since this parabola opens downwards (the coefficient of 𝑥² is negative), it will be below the x-axis (negative) for all x-values except at x = 3. Therefore, the entire range of x satisfies the inequality, and the solution to this problem is x < 3.
Examples & Analogies
Imagine a basketball thrown into the air. The parabola represents the ball's height. The ball is above the ground (height > 0) until it reaches a certain height at the peak (where the parabola touches zero). Below that height and as it falls back down, the ball is in a position where it is under 0 height, which corresponds to where our inequality holds.
Practice Problem 4
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- Solve and represent on a number line: 3𝑥² + 𝑥 - 2 ≥ 0
Detailed Explanation
To solve the inequality 3𝑥² + 𝑥 - 2 ≥ 0, we first equate to zero: 3𝑥² + 𝑥 - 2 = 0. We can apply the quadratic formula 𝑥 = [-𝑏 ± √(𝑏² - 4𝑎𝑐)]/(2𝑎). Here, 𝑎 = 3, 𝑏 = 1, and 𝑐 = -2. This yields roots of approximately 𝑥 = 0.67 and 𝑥 = -1. We examine intervals divided by these roots, checking test points from each interval. After testing values, the valid intervals where the expression is greater than or equal to zero will be combined and represented on a number line as part of the final answer.
Examples & Analogies
Consider the trajectory of a paper airplane you've thrown. There will be times (intervals) during its flight when it is above a designated benchmark height (say the table). Just as the path of the airplane crosses above and below this benchmark, solving the inequality will tell us during which flight intervals it's soaring high enough.
Practice Problem 5
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- A ball is thrown and its height (in meters) is modeled by ℎ(𝑡) = -5𝑡² + 20𝑡. For how long is the height at least 15 meters?
Detailed Explanation
To find out the time t when the height ℎ(𝑡) is at least 15 meters, we set up the inequality -5𝑡² + 20𝑡 ≥ 15 and rearrange it to standard form by moving 15 to the left side: -5𝑡² + 20𝑡 - 15 ≥ 0. Dividing through by -5 (and switching the inequality sign), we have 𝑡² - 4𝑡 + 3 ≤ 0. Factor this as (𝑡 - 1)(𝑡 - 3) ≤ 0. We find roots at 𝑡 = 1 and 𝑡 = 3, which divides the time line into intervals. After testing the intervals, we will conclude that t is valid between 1 and 3 seconds, which means the ball is at least 15 meters high during this time frame.
Examples & Analogies
Imagine you're timing an amusement park ride where the height of the ride (like a roller coaster) needs to be at least 15 meters to thrill the passengers. Just as the ride goes up and down during its journey, knowing the time intervals when it's above that height helps you better understand the experience.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Quadratic Expression: An expression in the form of 𝑎𝑥² + 𝑏𝑥 + 𝑐.
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Quadratic Inequality: Inequalities that involve quadratic expressions.
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Interval Notation: A way to represent solution sets for inequalities.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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Example of solving 𝑥² + 2𝑥 − 15 > 0 by factoring.
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Example of applying quadratic inequalities to projectile motion.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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Solve the parabola, find the roots, check the signs, it's how you compute!
📖 Fascinating Stories
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Imagine throwing a ball high; knowing its peak is key to why.
🧠 Other Memory Gems
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RUSSS: Roots, Upward/Downward, Sign Test, Solution Set.
🎯 Super Acronyms
FASST
- Factor
- Analyze
- Solve
- Sign
- Test.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Quadratic Expression
Definition:
An algebraic expression of the form 𝑎𝑥² + 𝑏𝑥 + 𝑐, where 𝑎, 𝑏, and 𝑐 are real numbers and 𝑎 ≠ 0.
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Term: Quadratic Inequality
Definition:
An inequality involving a quadratic expression that takes forms like 𝑎𝑥² + 𝑏𝑥 + 𝑐 < 0.
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Term: Roots
Definition:
The solutions of the quadratic equation, given by 𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0.