Solve The Following Quadratic Inequality. Write Your Answer In Interval Notation.${ 6t^2 - 7t - 5 \leq 0 }$Answer: _______________Fill In The Blank With Your Answer.

Mar 9, 2025 by ADMIN 167 views

Solving Quadratic Inequalities: A Step-by-Step Guide

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Introduction

Quadratic inequalities are a type of mathematical expression that involves a quadratic function and an inequality sign. They are commonly used in various fields, including physics, engineering, and economics. In this article, we will focus on solving quadratic inequalities of the form $ax^2 + bx + c \leq 0$ , where $a$ , $b$ , and $c$ are constants.

Understanding Quadratic Inequalities

A quadratic inequality is a mathematical expression that involves a quadratic function and an inequality sign. The general form of a quadratic inequality is $ax^2 + bx + c \leq 0$ , where $a$ , $b$ , and $c$ are constants. The inequality sign can be either less than or equal to ( $\leq$ ) or greater than or equal to ( $\geq$ ).

Solving Quadratic Inequalities

To solve a quadratic inequality, we need to find the values of $x$ that satisfy the inequality. We can do this by factoring the quadratic expression, using the quadratic formula, or graphing the quadratic function.

Factoring Quadratic Expressions

One way to solve a quadratic inequality is to factor the quadratic expression. If the quadratic expression can be factored, we can set each factor equal to zero and solve for $x$ . This will give us the values of $x$ that make the quadratic expression equal to zero.

For example, consider the quadratic inequality $6t^2 - 7t - 5 \leq 0$ . We can factor the quadratic expression as follows:

6t^2 - 7t - 5 = (3t + 1)(2t - 5)

Now, we can set each factor equal to zero and solve for $t$ :

3t + 1 = 0 \Rightarrow t = -\frac{1}{3}

2t - 5 = 0 \Rightarrow t = \frac{5}{2}

Using the Quadratic Formula

Another way to solve a quadratic inequality is to use the quadratic formula. The quadratic formula is given by:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

We can use the quadratic formula to find the values of $x$ that satisfy the quadratic inequality.

For example, consider the quadratic inequality $6t^2 - 7t - 5 \leq 0$ . We can use the quadratic formula as follows:

x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(6)(-5)}}{2(6)}

x = \frac{7 \pm \sqrt{49 + 120}}{12}

x = \frac{7 \pm \sqrt{169}}{12}

x = \frac{7 \pm 13}{12}

Graphing Quadratic Functions

Another way to solve a quadratic inequality is to graph the quadratic function. We can graph the quadratic function by plotting the points on the coordinate plane and drawing a smooth curve through the points.

For example, consider the quadratic inequality $6t^2 - 7t - 5 \leq 0$ . We can graph the quadratic function as follows:

The graph of the quadratic function is a parabola that opens upward. The parabola intersects the x-axis at two points: $t = -\frac{1}{3}$ and $t = \frac{5}{2}$ . The parabola is below the x-axis between these two points, so the solution to the inequality is $-\frac{1}{3} \leq t \leq \frac{5}{2}$ .

Conclusion

In conclusion, solving quadratic inequalities involves finding the values of $x$ that satisfy the inequality. We can do this by factoring the quadratic expression, using the quadratic formula, or graphing the quadratic function. By following these steps, we can solve quadratic inequalities and find the solution in interval notation.

Example Solution

Let's consider the quadratic inequality $6t^2 - 7t - 5 \leq 0$ . We can solve this inequality by factoring the quadratic expression, using the quadratic formula, or graphing the quadratic function.

Factoring the Quadratic Expression

We can factor the quadratic expression as follows:

6t^2 - 7t - 5 = (3t + 1)(2t - 5)

Now, we can set each factor equal to zero and solve for $t$ :

3t + 1 = 0 \Rightarrow t = -\frac{1}{3}

2t - 5 = 0 \Rightarrow t = \frac{5}{2}

Using the Quadratic Formula

We can use the quadratic formula as follows:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(6)(-5)}}{2(6)}

x = \frac{7 \pm \sqrt{49 + 120}}{12}

x = \frac{7 \pm \sqrt{169}}{12}

Graphing the Quadratic Function

We can graph the quadratic function as follows:

Final Answer

The final answer to the quadratic inequality $6t^2 - 7t - 5 \leq 0$ is $-\frac{1}{3} \leq t \leq \frac{5}{2}$ .

References

[1] "Quadratic Inequalities" by Math Open Reference
[2] "Solving Quadratic Inequalities" by Khan Academy
[3] "Quadratic Formula" by Wolfram MathWorld

Keywords

Quadratic inequalities
Factoring quadratic expressions
Quadratic formula
Graphing quadratic functions
Interval notation
Solution to quadratic inequality
Quadratic inequality solution
Quadratic inequality examples
Quadratic inequality problems
Quadratic inequality solutions
Quadratic inequality examples with solutions
Quadratic inequality problems with solutions

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Introduction

Quadratic inequalities are a type of mathematical expression that involves a quadratic function and an inequality sign. They are commonly used in various fields, including physics, engineering, and economics. In this article, we will answer some of the most frequently asked questions about quadratic inequalities.

Q&A

Q: What is a quadratic inequality?

A: A quadratic inequality is a mathematical expression that involves a quadratic function and an inequality sign. The general form of a quadratic inequality is $ax^2 + bx + c \leq 0$ , where $a$ , $b$ , and $c$ are constants.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you can use one of the following methods:

Factoring the quadratic expression
Using the quadratic formula
Graphing the quadratic function

Q: What is the difference between a quadratic equation and a quadratic inequality?

A: A quadratic equation is a mathematical expression that involves a quadratic function and an equal sign, whereas a quadratic inequality is a mathematical expression that involves a quadratic function and an inequality sign.

Q: Can I use the quadratic formula to solve a quadratic inequality?

A: Yes, you can use the quadratic formula to solve a quadratic inequality. However, you need to be careful when using the quadratic formula, as it may not always give you the correct solution.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you can use a graphing calculator or a computer program. You can also plot the points on the coordinate plane and draw a smooth curve through the points.

Q: What is the solution to a quadratic inequality?

A: The solution to a quadratic inequality is the set of values of $x$ that satisfy the inequality. It can be expressed in interval notation, such as $[a, b]$ or $(-\infty, a] \cup [b, \infty)$ .

Q: Can I use a calculator to solve a quadratic inequality?

A: Yes, you can use a calculator to solve a quadratic inequality. Many calculators have a built-in function for solving quadratic inequalities.

Q: How do I check my solution to a quadratic inequality?

A: To check your solution to a quadratic inequality, you can plug in a value of $x$ that is in the solution set and verify that the inequality is true.

Q: Can I use a quadratic inequality to model real-world problems?

A: Yes, you can use a quadratic inequality to model real-world problems. Quadratic inequalities are commonly used in physics, engineering, and economics to model problems such as projectile motion, optimization, and resource allocation.

Example Solutions

Example 1: Solving a Quadratic Inequality using Factoring

Consider the quadratic inequality $6t^2 - 7t - 5 \leq 0$ . We can factor the quadratic expression as follows:

6t^2 - 7t - 5 = (3t + 1)(2t - 5)

Now, we can set each factor equal to zero and solve for $t$ :

3t + 1 = 0 \Rightarrow t = -\frac{1}{3}

2t - 5 = 0 \Rightarrow t = \frac{5}{2}

The solution to the inequality is $-\frac{1}{3} \leq t \leq \frac{5}{2}$ .

Example 2: Solving a Quadratic Inequality using the Quadratic Formula

Consider the quadratic inequality $2x^2 + 5x + 3 \leq 0$ . We can use the quadratic formula as follows:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

x = \frac{-5 \pm \sqrt{5^2 - 4(2)(3)}}{2(2)}

x = \frac{-5 \pm \sqrt{25 - 24}}{4}

x = \frac{-5 \pm \sqrt{1}}{4}

The solution to the inequality is $x = -\frac{1}{2}$ .

Example 3: Solving a Quadratic Inequality using Graphing

Consider the quadratic inequality $x^2 + 4x + 4 \leq 0$ . We can graph the quadratic function as follows:

The graph of the quadratic function is a parabola that opens upward. The parabola intersects the x-axis at one point: $x = -2$ . The parabola is below the x-axis between this point and the origin, so the solution to the inequality is $-2 \leq x \leq 0$ .

Conclusion

In conclusion, quadratic inequalities are a type of mathematical expression that involves a quadratic function and an inequality sign. They are commonly used in various fields, including physics, engineering, and economics. By following the steps outlined in this article, you can solve quadratic inequalities and find the solution in interval notation.

References

[1] "Quadratic Inequalities" by Math Open Reference
[2] "Solving Quadratic Inequalities" by Khan Academy
[3] "Quadratic Formula" by Wolfram MathWorld

Keywords

Quadratic inequalities
Factoring quadratic expressions
Quadratic formula
Graphing quadratic functions
Interval notation
Solution to quadratic inequality
Quadratic inequality solution
Quadratic inequality examples
Quadratic inequality problems
Quadratic inequality solutions
Quadratic inequality examples with solutions
Quadratic inequality problems with solutions