Solve The Nonlinear Inequality. Express The Solution Using Interval Notation. { (x+4)(x-7)(x-8) \leq 0$}$Graph The Solution Set.

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Introduction

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Nonlinear inequalities are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving the nonlinear inequality $(x+4)(x-7)(x-8) \leq 0$ and express the solution using interval notation. We will also graph the solution set to provide a visual representation of the solution.

Understanding Nonlinear Inequalities

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Nonlinear inequalities are inequalities that involve a nonlinear expression, such as a polynomial or a rational expression. They can be written in the form $f(x) \leq 0$ or $f(x) \geq 0$, where $f(x)$ is a nonlinear function. Nonlinear inequalities can be solved using various methods, including factoring, graphing, and numerical methods.

Factoring the Nonlinear Inequality

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To solve the nonlinear inequality $(x+4)(x-7)(x-8) \leq 0$, we can start by factoring the expression. Factoring involves expressing the expression as a product of simpler expressions, called factors. In this case, we can factor the expression as follows:

$$(x+4)(x-7)(x-8) = (x+4)(x-8)(x-7)$$

Finding the Critical Points

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The critical points of the nonlinear inequality are the values of $x$ that make the expression equal to zero. In this case, the critical points are $x=-4$, $x=7$, and $x=8$. These points divide the number line into four intervals: $(-\infty, -4)$, $(-4, 7)$, $(7, 8)$, and $(8, \infty)$.

Testing the Intervals

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To determine which intervals satisfy the inequality, we can test each interval by substituting a test value into the expression. Let's test each interval:

Interval $(-\infty, -4)$

Substituting $x=-5$ into the expression, we get:

$$(x+4)(x-7)(x-8) = (-5+4)(-5-7)(-5-8) = (-1)(-12)(-13) = -156$$

Since the expression is negative, the interval $(-\infty, -4)$ satisfies the inequality.

Interval $(-4, 7)$

Substituting $x=0$ into the expression, we get:

$$(x+4)(x-7)(x-8) = (0+4)(0-7)(0-8) = (4)(-7)(-8) = 224$$

Since the expression is positive, the interval $(-4, 7)$ does not satisfy the inequality.

Interval $(7, 8)$

Substituting $x=7.5$ into the expression, we get:

$$(x+4)(x-7)(x-8) = (7.5+4)(7.5-7)(7.5-8) = (11.5)(0.5)(-0.5) = -2.875$$

Since the expression is negative, the interval $(7, 8)$ satisfies the inequality.

Interval $(8, \infty)$

Substituting $x=9$ into the expression, we get:

$$(x+4)(x-7)(x-8) = (9+4)(9-7)(9-8) = (13)(2)(1) = 26$$

Since the expression is positive, the interval $(8, \infty)$ does not satisfy the inequality.

Expressing the Solution in Interval Notation

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Based on the results of the interval testing, we can express the solution in interval notation as follows:

$$(-\infty, -4) \cup (7, 8)$$

This notation indicates that the solution set consists of two intervals: $(-\infty, -4)$ and $(7, 8)$.

Graphing the Solution Set

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To graph the solution set, we can plot the critical points on the number line and shade the intervals that satisfy the inequality. The graph will consist of two intervals: $(-\infty, -4)$ and $(7, 8)$.

Conclusion

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Solving nonlinear inequalities requires a thorough understanding of the concept and the ability to apply various methods, including factoring, graphing, and numerical methods. In this article, we solved the nonlinear inequality $(x+4)(x-7)(x-8) \leq 0$ and expressed the solution using interval notation. We also graphed the solution set to provide a visual representation of the solution.

Frequently Asked Questions

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Q: What is a nonlinear inequality?

A: A nonlinear inequality is an inequality that involves a nonlinear expression, such as a polynomial or a rational expression.

Q: How do I solve a nonlinear inequality?

A: To solve a nonlinear inequality, you can use various methods, including factoring, graphing, and numerical methods.

Q: What is interval notation?

A: Interval notation is a way of expressing a solution set using intervals on the number line.

Q: How do I graph a solution set?

A: To graph a solution set, you can plot the critical points on the number line and shade the intervals that satisfy the inequality.

References

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• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by James Stewart
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Keywords

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• Nonlinear inequality
• Interval notation
• Graphing
• Factoring
• Numerical methods
• Algebra
• Trigonometry
• Calculus
• Mathematics
• Computer science
  

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Introduction

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Nonlinear inequalities are a fundamental concept in mathematics, and solving them can be a challenging task. In this article, we will address some of the most frequently asked questions about nonlinear inequalities, including how to solve them, how to express the solution in interval notation, and how to graph the solution set.

Q&A

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Q: What is a nonlinear inequality?

A: A nonlinear inequality is an inequality that involves a nonlinear expression, such as a polynomial or a rational expression.

Q: How do I solve a nonlinear inequality?

A: To solve a nonlinear inequality, you can use various methods, including factoring, graphing, and numerical methods. Factoring involves expressing the expression as a product of simpler expressions, called factors. Graphing involves plotting the expression on a graph and identifying the intervals that satisfy the inequality. Numerical methods involve using a calculator or computer to approximate the solution.

Q: What is interval notation?

A: Interval notation is a way of expressing a solution set using intervals on the number line. It is a shorthand way of writing the solution set, and it can be used to express the solution in a more concise and elegant way.

Q: How do I express the solution in interval notation?

A: To express the solution in interval notation, you need to identify the intervals that satisfy the inequality. You can do this by testing each interval by substituting a test value into the expression. If the expression is negative, the interval satisfies the inequality. If the expression is positive, the interval does not satisfy the inequality.

Q: How do I graph a solution set?

A: To graph a solution set, you need to plot the critical points on the number line and shade the intervals that satisfy the inequality. The critical points are the values of x that make the expression equal to zero. You can use a graphing calculator or computer to plot the graph.

Q: What is the difference between a linear inequality and a nonlinear inequality?

A: A linear inequality is an inequality that involves a linear expression, such as a polynomial of degree one. A nonlinear inequality is an inequality that involves a nonlinear expression, such as a polynomial of degree two or higher.

Q: How do I determine the number of solutions to a nonlinear inequality?

A: To determine the number of solutions to a nonlinear inequality, you need to examine the graph of the expression. If the graph has no x-intercepts, the inequality has no solutions. If the graph has one x-intercept, the inequality has one solution. If the graph has two x-intercepts, the inequality has two solutions.

Q: Can I use a calculator or computer to solve a nonlinear inequality?

A: Yes, you can use a calculator or computer to solve a nonlinear inequality. Many graphing calculators and computer algebra systems have built-in functions that can solve nonlinear inequalities.

Q: How do I check my work when solving a nonlinear inequality?

A: To check your work, you need to substitute a test value into the expression and verify that the inequality is satisfied. You can also use a graphing calculator or computer to plot the graph and verify that the solution set is correct.

Conclusion

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Nonlinear inequalities are a fundamental concept in mathematics, and solving them can be a challenging task. By understanding the different methods for solving nonlinear inequalities, including factoring, graphing, and numerical methods, you can develop the skills and confidence you need to tackle even the most difficult problems. Remember to always check your work and use a graphing calculator or computer to verify your solution.

Frequently Asked Questions

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Q: What is the difference between a rational inequality and a nonlinear inequality?

A: A rational inequality is an inequality that involves a rational expression, such as a fraction. A nonlinear inequality is an inequality that involves a nonlinear expression, such as a polynomial of degree two or higher.

Q: How do I solve a rational inequality?

A: To solve a rational inequality, you need to factor the numerator and denominator, and then use the factored form to identify the critical points. You can then use the critical points to determine the solution set.

Q: What is the difference between a polynomial inequality and a nonlinear inequality?

A: A polynomial inequality is an inequality that involves a polynomial expression, such as a polynomial of degree two or higher. A nonlinear inequality is an inequality that involves a nonlinear expression, such as a polynomial of degree two or higher.

Q: How do I solve a polynomial inequality?

A: To solve a polynomial inequality, you need to factor the polynomial, and then use the factored form to identify the critical points. You can then use the critical points to determine the solution set.

References

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• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by James Stewart
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Keywords

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• Nonlinear inequality
• Interval notation
• Graphing
• Factoring
• Numerical methods
• Algebra
• Trigonometry
• Calculus
• Mathematics
• Computer science
• Rational inequality
• Polynomial inequality