Quadratic Inequalities PDF

6 Quadratic Inequalities Prepared By: Mr. Jonathan Gumapac, Mr. Noel Juan, Ms. Annabelle Petate Objectives The learner will be able to 1. illustrate quadratic Business Decision-Making: Beyo...

6 Quadratic Inequalities Prepared By: Mr. Jonathan Gumapac, Mr. Noel Juan, Ms. Annabelle Petate Objectives The learner will be able to 1. illustrate quadratic Business Decision-Making: Beyond Exact Values inequalities, In our everyday lives, we are 2. solve quadratic surrounded by a vast number of inequalities, and mathematical concepts. We may not even be aware that we are applying 3. solve problems them in various aspects of our daily involving quadratic routines. In our previous lessons, we inequalities. frequently encountered terms like "equal," "equate," and "equation," all of which indicate that we are looking for precise values. However, real life is not limited to these concepts alone. What if we find ourselves facing situations where we need to estimate or work with value ranges? For instance, consider the role of a businessperson. Setting clear goals is a Curiosity Strike! fundamental quality in Can you think of business management. Yet, other situations in the world of business there where inequality are full of uncertainties, can be applied? setting extremely exact goals can be a challenge. Instead, we often find ourselves setting goals within a range, such as aiming for a profit greater than a specific value. This is where the concept of quadratic inequality takes place. It can be an invaluable tool to make favorable decisions. Whether it involves estimating potential outcomes or establishing flexible objectives. In this module, we will learn how deal with problems involving quadratic inequalities and how useful it can be in different fields. Page ⎮1 Properties of Inequality The following properties of inequalities are useful in the solution of the quadratic inequality: i. A product is positive when the factors are both positive, or both negative. That is, if 𝐴𝐵 > 0, then 𝐴 > 0 and 𝐵 > 0, or 𝐴 < 0 and 𝐵 < 0. ii. The product is negative when one factor is positive and the other is negative.. That is, if 𝐴𝐵 < 0, then 𝐴 > 0 and 𝐵 < 0, or 𝐴 < 0 and 𝐵 > 0. SOLVING QUADRATIC INEQUALITIES ALGEBRAICALLY The algebraic method we will be using requires your knowledge in solving quadratic equations. We will find the critical points for the inequality, which will be the solutions to the related quadratic equation. Then, we will use the critical points to divide the number line into intervals and then determine whether the quadratic expression will be positive or negative in the interval. We then determine the solution for the inequality. Example 1. Solve 𝒙𝟐 − 𝟔𝒙 ≥ 𝟏𝟔. Write the solution in interval notation. Solution: Page ⎮2 Study Tip Factor the trinomial, then use the properties of inequality to determine the correct interval(s). Page ⎮3  Example 2. Solve −𝒙𝟐 − 𝟑𝒙 + 𝟏𝟖 > 𝟎. Write the solution in interval notation. Solution: Study Tip When you divide all the terms in a quadratic inequality by a negative number, the direction of the inequality symbol is reversed. Page ⎮4 Page ⎮5  Example 3. Solve 𝟑𝒙𝟐 − 𝟖𝒙 + 𝟒 ≥ 𝟎. Write the solution in interval notation. Solution: Study Tip Factor the trinomial, then use the properties of inequality to determine the correct interval(s). Page ⎮6  Solving Problems Involving Quadratic Inequalities Example 4 Mr. Fiero is planning to construct a nipa hut. The rectangular area’s length is 5 m longer than its width. If he plans to have an area that is less than or equal to 414 𝑚 2, then what should be the possible values of the width? Study Tip When you divide all the terms in a quadratic inequality by a negative number, the direction of the inequality symbol is reversed. Page ⎮7 Page ⎮8  Example 5 An object is launched upward at 80 feet per second from a platform 80 feet high. This can be modeled by the function ℎ = −16𝑡 2 + 80𝑡 + 80, where 𝑡, is time in seconds and h is the height. What time will the object reach a height that is greater than 176 ft. Study Tip When you divide all the terms in a quadratic inequality by a negative number, the direction of the inequality symbol is reversed. Page ⎮9  Example 6 Study Tip The profit of a candy shop business can be modeled by the profit function 𝑃(𝑥) = −28𝑥 2 − 182𝑥 + 336, where 𝑃(𝑥) represents the profit in When you divide all pesos, and 𝑥 is the price of the item. What minimum price at which the the terms in a owner sell her candies to ensure a profit greater than 0? quadratic inequality Solution by a negative number, the direction of the inequality symbol is reversed. Page ⎮10 Page ⎮11 Practice Exercises: A. Solve the following quadratic inequalities. Write your answer in interval notation. 1. 𝑥 2 − 4𝑥 < 5 2. 2𝑥 2 − 3𝑥 ≥ 2 3. −𝑥 2 + 6𝑥 − 9 < 0 4. 3𝑥 2 + 2𝑥 > 5𝑥 + 6 5. 𝑥 2 + 8𝑥 + 12 ≥ 0 B. Solve the following problem. 6. A rocket is launched from the ground. Its height ℎ(𝑡) above the ground in meters is modeled by the quadratic function ℎ(𝑡) = −5𝑡 2 + 40𝑡, where 𝑡 is the time in seconds. Find the time interval during which the rocket's height is greater than 30 meters. 7. A small bakery produces and sells cupcakes. The profit function for the bakery is given by 𝑃(𝑥) = −2𝑥 2 + 50𝑥 − 100, where 𝑃(𝑥) represents the profit in dollars and 𝑥 is the number of cupcakes produced and sold. The bakery owner wants to know how many cupcakes should be sold to ensure a profit of at least $200. Determine the minimum number of cupcakes that need to be sold to achieve this goal. Page ⎮12 Page ⎮13 Effectrive Goal Setting using the Concept of Quadratic Inequality As a business goes and grows over the year, one can predict its potential profit by analyzing the behavior of the number of quantity being sold and the price being set. This helps a lot of big businesses to foresee opportunities in their business. All these behaviors of price and quantity can be mathematically modeled through profit function. A profit function establishes a mathematical connection between a company's overall profit and its production. This function is calculated as total revenue minus total costs. Once a mathematical model of the profit is written, they can use this to weigh situations. This means that a business owner can see the potential effects of the sudden change in price or production to the profit. This can also be used to set profit goals. For example, a profit function of 𝑃(𝑥) = −0.5𝑥 2 + 60𝑥 − 10,000 , and you want to see how many items you need to produce to have a profit of greater than ₱15,000. The concept of solving quadratic inequalities that we discussed in this module can be applied to identify the range of number that can help us achieve our goal of ₱15,000. Page ⎮14  It’s now time to write your self evaluation in this lesson. Please provide a feedback on where you are with your learning for this lesson. References Orines, Fernando B. et. al.(2018). Next Century Mathematics. Quezon City: Phoenix Publishing House. Writing and Using Inequalitites https://content.nroc.org/Algebra.HTML5/U05L1T3/TopicText/en/text.html Optimisation (Business) https://www.ncl.ac.uk/webtemplate/ask-assets/external/maths- resources/business/differentiation-and-optimisation/optimisation.html Juan, Noel S., Agao, Christian Kurt V., Aboleda, Sonia D., Elauria, Wilson B. (2021). Solving Quadratic Equations.[Unpublished manuscript]. Mathematics Area, Junior High School Division, DLSU Integrated School. Juan, Noel S., Gumapac, Jonathan F., Aboleda, Sonia D. (2022). Solving Quadratic Inequality.[Unpublished manuscript]. Mathematics Area, Junior High School Division, DLSU Integrated School. Page ⎮15 Page ⎮16