What Is the Solution Set of the Quadratic Inequality mc008-1.jpg

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What Is the Solution Set of the Quadratic Inequality mc008-1.jpg?

Quadratic inequalities are a type of mathematical expression that involve a quadratic function and an inequality sign. The solution set of a quadratic inequality represents the values of the variable that satisfy the given inequality. In this article, we will discuss how to find the solution set of the quadratic inequality mc008-1.jpg and provide examples to illustrate the concept.

To determine the solution set of a quadratic inequality, we need to consider two factors: the sign of the quadratic function and the inequality sign. The quadratic function can be written in the form ax^2 + bx + c, where a, b, and c are constants. The inequality sign can be either greater than (>), greater than or equal to (≥), less than (<), or less than or equal to (≤). Let’s consider the quadratic inequality mc008-1.jpg as an example. To find the solution set, we will follow these steps: Step 1: Solve the quadratic equation mc008-1.jpg = 0 to find the x-intercepts or roots of the equation. The roots of the quadratic equation represent the points where the quadratic function intersects the x-axis. Step 2: Plot the roots on a number line and mark them accordingly. Step 3: Determine the sign of the quadratic function in the intervals between the roots. To do this, choose a test point within each interval and substitute it into the quadratic function. If the resulting value is positive, the quadratic function is positive in that interval. If it is negative, the quadratic function is negative in that interval.

Step 4: Determine the sign of the quadratic inequality. Depending on the inequality sign, the solution set will be either the intervals where the quadratic function is positive or negative, or the union of these intervals. Let’s apply these steps to the quadratic inequality mc008-1.jpg. The first step is to solve the quadratic equation mc008-1.jpg = 0. By factoring or using the quadratic formula, we find the roots to be x = 1 and x = 4. Next, we plot the roots on a number line: 1 4 Now, we choose a test point within each interval and substitute it into the quadratic function. Let’s choose x = 0 and x = 5 as test points. Substituting x = 0 into mc008-1.jpg, we get f(0) = -12, which is negative. Substituting x = 5 into mc008-1.jpg, we get f(5) = 80, which is positive. Based on these results, we can determine the sign of the quadratic function in each interval: Interval (-∞, 1): f(x) < 0
Interval (1, 4): f(x) > 0
Interval (4, ∞): f(x) < 0 Next, we consider the inequality sign in mc008-1.jpg, which is greater than or equal to (≥). This means that the solution set includes the intervals where the quadratic function is greater than or equal to zero. Therefore, the solution set of the quadratic inequality mc008-1.jpg is [1, 4] or the closed interval from 1 to 4. This means that all values of x between 1 and 4, including 1 and 4, satisfy the given inequality. FAQs: Q: What if the inequality sign is less than or equal to (≤)?
A: If the inequality sign is ≤, the solution set includes the intervals where the quadratic function is less than or equal to zero.

Q: What if the quadratic function cannot be factored or solved using the quadratic formula?
A: In such cases, you can use graphical methods or approximation techniques to determine the approximate solution set.

Q: How can I check if my solution set is correct?
A: You can substitute different values of x from the solution set into the quadratic inequality and check if the inequality holds true.

Q: Can a quadratic inequality have multiple solution sets?
A: Yes, a quadratic inequality can have multiple solution sets if it involves absolute value or if the quadratic function intersects the x-axis at multiple points.

In conclusion, the solution set of the quadratic inequality mc008-1.jpg is the closed interval [1, 4]. By following the steps mentioned above, you can find the solution set of any quadratic inequality. Remember to consider the sign of the quadratic function and the inequality sign to determine the intervals that satisfy the inequality.
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