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Chapter 0: Problem 126

Describe ways in which solving a linear inequality is different than solving a linear equation.

Short Answer

Expert verified
The methods for solving linear equations and inequalities differ in how you handle the inequality sign. For equations, you perform the same operation on both sides, while for inequalities, if you multiply or divide by a negative number, the direction of the inequality changes. Also, the solution to a linear equation is a specific point, but the solution to a linear inequality is a range of values.

Step by step solution

01

Define Linear Equation

A linear equation is an equation for a straight line. It is called 'linear' because the graph that represents it is a straight line. The standard form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept, the point where the line crosses the y-axis. To solve a linear equation, find the value of the variable that makes the equation true.
02

Define Linear Inequality

A linear inequality is similar to a linear equation but instead of an equals sign, it has an inequality sign (> , < , \鈮� , \鈮�). The linear inequality describes a region of the graph and not just a line. It can represent a range of solutions. When solving a linear inequality, it's necessary to find all values of the variable that make the inequality true.
03

Explain the Difference in Method

The main difference in the methods for solving linear equations and inequalities lies in how the inequality sign is handled. For an equation, you perform the same operation on both sides to maintain the equality. For an inequality, if you multiply or divide by a negative number, the direction of the inequality changes. This is a crucial difference between the two.
04

Explain the Difference in the Result

The solution of a linear equation is a specific point or points where the line crosses the x-axis. The solution to a linear inequality, however, is a range of values. The graph of this solution will shading a region of the graph that represents all the solutions to the inequality.

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Most popular questions from this chapter

Will help you prepare for the material covered in the first section of the next chapter. If \(y=4-x,\) find the value of \(y\) that corresponds to values of \(x\) for each integer starting with -3 and ending with 3

What's wrong with this argument? Suppose \(x\) and \(y\) represent two real numbers, where \(x>y .\) $$\begin{aligned} &2>1\\\ &2(y-x)>1(y-x)\\\ &2 y-2 x>y-x\\\ &\begin{aligned} y-2 x &>-x \\ y &>x \end{aligned} \end{aligned}$$ The final inequality, \(y>x,\) is impossible because we were initially given \(x>y\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$6+\frac{1}{x}=\frac{7}{x}$$

Perform the indicated operations. Simplify the result, if possible. $$\frac{a b}{a^{2}+a b+b^{2}}+\left(\frac{a c-a d-b c+b d}{a c-a d+b c-b d} \div \frac{a^{3}-b^{3}}{a^{3}+b^{3}}\right)$$

Rationalize the numerator. $$\frac{\sqrt{x}+\sqrt{y}}{x^{2}-y^{2}}$$
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