/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 Fill in the blanks. \(y \leq x... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 3

Fill in the blanks. \(y \leq x^{2}-4 x+3\) is an example of a nonlinear inequality in ____ variables.

Short Answer

Expert verified
two

Step by step solution

01

Identify the Type of Equation

Examine the given inequality \(y \leq x^2 - 4x + 3\). Notice that the equation involves both an \(x\) term and a \(y\) term. The squared \(x^2\) indicates it is a quadratic expression.
02

Determine the Unknowns

Count the number of distinct variables in the inequality. The expression includes two distinct variables: \(x\) and \(y\).
03

Classify the Inequality

Since there are two variables, \(x\) and \(y\), and the inequality relates them, it is a nonlinear inequality in two variables.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Expression
A quadratic expression is a type of polynomial where the highest degree of a variable is 2. This means that the variable is squared, or raised to the power of 2. In the given inequality, the expression is \(x^2 - 4x + 3\), demonstrating its quadratic nature because it contains the \(x^2\) term. Quadratic expressions are fundamental in understanding various algebraic concepts because they represent parabolic graphs when plotted on a coordinate plane.

Key characteristics of a quadratic expression include:
  • The presence of a square term, \(x^2\).
  • It may also have linear \(x\) and constant terms, such as \(-4x\) and \(+3\) respectively in our example.
  • Quadratic expressions always result in a curve known as a parabola.
Understanding these characteristics helps in solving and graphing inequalities that involve quadratic expressions.
Two Variables
In mathematics, when we discuss equations or inequalities involving two variables, we usually refer to relationships between two distinct unknowns. For example, in the inequality \(y \leq x^2 - 4x + 3\), the variables are \(x\) and \(y\). These variables generally represent two different quantities that may change depending on the scenario.

In this context:
  • \(x\) is often considered the independent variable, which means its value is freely chosen.
  • \(y\) is considered the dependent variable, as its value depends on \(x\).
Working with two variables allows us to model complex relationships and represent them graphically on the Cartesian plane. This can help visualize solutions or understand different aspects of the model formed by the inequality.
Mathematical Inequality
A mathematical inequality expresses the relationship between two expressions that are not strictly equal. Instead of an equals sign, inequalities use symbols like \(\leq\), \(<\), \(\geq\), or \(>\) to denote that one side is either less than, greater than, or equal but not identical to the other side. In our example, \(y \leq x^2 - 4x + 3\) is an inequality, indicating that \(y\) is less than or equal to the quadratic expression.

The key components of dealing with inequalities are:
  • Understanding the direction of the inequality symbol, which influences the solutions you seek.
  • Finding the solution set, which may include a range of values rather than specific points, showing all potential \(x\) and \(y\) pairs that satisfy the inequality.
  • Graphically representing these solutions can lead to a shaded region on a graph, illustrating all combinations that make the inequality true.
Overall, mastering inequalities empowers you to solve problems involving limits and bounds, contributing deeply to mathematical analysis in real-world applications.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Solve each equation. Approximate the solutions to the nearest hundredth when appropriate. $$x(x-27)=280$$

Solve each equation. Approximate the solutions to the nearest hundredth when appropriate. $$2 x^{2}+3 x=-3$$

Simplify each expression. Assume all variables represent positive numbers. $$ 3(\sqrt{5 b}-\sqrt{3})^{2} $$

Let \(g(x)=4.5 x^{2}+0.2 x .\) For what value(s) of \(x\) is \(g(x)=3.75 ?\)

Solve each equation. Approximate the solutions to the nearest hundredth when appropriate. $$\frac{x^{2}}{8}-\frac{x}{4}=\frac{1}{2}$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.