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Questions and Answers
What is the primary difference between equations and inequalities?
What is the primary difference between equations and inequalities?
In the context of business, why might using inequalities be more practical than using equations?
In the context of business, why might using inequalities be more practical than using equations?
Which of these scenarios is best represented by a quadratic inequality?
Which of these scenarios is best represented by a quadratic inequality?
What is a key attribute of a businessperson that supports the use of inequalities?
What is a key attribute of a businessperson that supports the use of inequalities?
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What does the module suggest is an 'invaluable tool' for business decision-making?
What does the module suggest is an 'invaluable tool' for business decision-making?
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What does a profit function mathematically connect?
What does a profit function mathematically connect?
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When might someone encounter the application of mathematical concepts?
When might someone encounter the application of mathematical concepts?
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How is a profit function typically calculated?
How is a profit function typically calculated?
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What mathematical concept is associated with terms like 'equal', 'equate', and 'equation'?
What mathematical concept is associated with terms like 'equal', 'equate', and 'equation'?
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Why is setting 'extremely exact goals' described as potentially challenging in the business world?
Why is setting 'extremely exact goals' described as potentially challenging in the business world?
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If a company's profit function is defined as $P(x) = -0.5x^2 + 60x - 10,000$, what does 'x' likely represent?
If a company's profit function is defined as $P(x) = -0.5x^2 + 60x - 10,000$, what does 'x' likely represent?
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What is one way a business owner might utilize their profit function?
What is one way a business owner might utilize their profit function?
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What mathematical concept is applied to find the range of production needed to surpass a specific profit goal using a profit function?
What mathematical concept is applied to find the range of production needed to surpass a specific profit goal using a profit function?
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If a company has a profit function and wants to achieve a profit of ₱15,000, what can this function help determine?
If a company has a profit function and wants to achieve a profit of ₱15,000, what can this function help determine?
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A business owner has a profit function $P(x)$.What action demonstrates using this profit function for business planning?
A business owner has a profit function $P(x)$.What action demonstrates using this profit function for business planning?
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A profit function is a mathematical way of expressing a company's financial performance. Which of the following can it directly help with?
A profit function is a mathematical way of expressing a company's financial performance. Which of the following can it directly help with?
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A projectile's height is modeled by $h = -16t^2 + 80t + 80$. What time interval will the projectile be above 176 feet?
A projectile's height is modeled by $h = -16t^2 + 80t + 80$. What time interval will the projectile be above 176 feet?
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If a candy shop's profit is modeled by $P(x) = -28x^2 - 182x + 336$, what is the minimum price to ensure a profit greater than 0?
If a candy shop's profit is modeled by $P(x) = -28x^2 - 182x + 336$, what is the minimum price to ensure a profit greater than 0?
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Solve the quadratic inequality $x^2 - 4x < 5$. Which interval notation represents the correct solution?
Solve the quadratic inequality $x^2 - 4x < 5$. Which interval notation represents the correct solution?
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Find a solution to the inequality $2x^2 - 3x \ge 2$ using interval notation.
Find a solution to the inequality $2x^2 - 3x \ge 2$ using interval notation.
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Which of the following interval notations represents the solution to $-x^2 + 6x - 9 < 0$?
Which of the following interval notations represents the solution to $-x^2 + 6x - 9 < 0$?
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What interval notates the solution for $3x^2 + 2x > 5x + 6$?
What interval notates the solution for $3x^2 + 2x > 5x + 6$?
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A rocket's height is modeled by $h(t) = -5t^2 + 40t$. What time interval is the rocket above 30 meters?
A rocket's height is modeled by $h(t) = -5t^2 + 40t$. What time interval is the rocket above 30 meters?
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A bakery's profit is given by $P(x) =-2x^2 + 50x - 100$. How many cupcakes must be sold to ensure a profit of at least $200?
A bakery's profit is given by $P(x) =-2x^2 + 50x - 100$. How many cupcakes must be sold to ensure a profit of at least $200?
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When is the product of two factors negative?
When is the product of two factors negative?
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What are 'critical points' in the context of solving quadratic inequalities?
What are 'critical points' in the context of solving quadratic inequalities?
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If a quadratic inequality is multiplied by a negative number, what happens to the inequality symbol?
If a quadratic inequality is multiplied by a negative number, what happens to the inequality symbol?
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Given the inequality $x^2 - 6x \geq 16$, what are the critical points?
Given the inequality $x^2 - 6x \geq 16$, what are the critical points?
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What is the solution set for the inequality $-x^2 - 3x + 18 > 0$?
What is the solution set for the inequality $-x^2 - 3x + 18 > 0$?
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What is the solution set for $3x^2 - 8x + 4 \geq 0$?
What is the solution set for $3x^2 - 8x + 4 \geq 0$?
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If the width of a rectangle is represented by w and the length is 5 meters longer, and the area is less than or equal to 414 $m^2$, which inequality represents this scenario?
If the width of a rectangle is represented by w and the length is 5 meters longer, and the area is less than or equal to 414 $m^2$, which inequality represents this scenario?
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An object is launched upwards at 80 feet per second from a 80-foot high platform. Which inequality can be used to find when the object is more than 144 feet above ground, using the height formula $h = -16t^2 + 80t + 80$?
An object is launched upwards at 80 feet per second from a 80-foot high platform. Which inequality can be used to find when the object is more than 144 feet above ground, using the height formula $h = -16t^2 + 80t + 80$?
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Study Notes
Quadratic Inequalities
- Quadratic inequalities are expressions involving quadratic equations with inequality symbols such as >, <, ≥, ≤, and ≠.
- They are expressions that can be expressed in various forms: ax² + bx + c > 0, ax² + bx + c < 0, ax² + bx + c ≥ 0, ax² + bx + c ≤ 0, and ax² + bx + c ≠ 0.
- A solution for a quadratic inequality is any set of x-values that make the inequality true.
- Useful properties for solving quadratic inequalities are: a product is positive when factors are either both positive or both negative; and a product is negative when the factors have opposite signs.
Solving Quadratic Inequalities Algebraically
- Algebraic methods involve using knowledge of quadratic equations to find critical points of the inequality.
- Critical points are the solutions to the related quadratic equation.
- The critical points divide the number line into intervals.
- Substitute test points from each interval into the original inequality to determine whether the inequality holds true.
- Use the intervals where the inequality is true to determine the solutions.
- Express solutions in interval notation.
Additional Notes
- Applying quadratic inequalities to solve real-world problems involves analyzing the problem to identify the relevant equation and inequality.
- Real-world problems often involve relationships, such as profit functions, where the goal is to determine the required quantity to achieve a certain outcome.
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Description
Test your understanding of quadratic inequalities and their properties in this quiz. You will learn to solve expressions involving inequality symbols and find critical points along with interval testing. Perfect for algebra students looking to reinforce their skills in this area.
