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Chapter 0: Problem 125

Describe ways in which solving a linear inequality is similar to solving a linear equation.

Short Answer

Expert verified
Solving a linear inequality is similar to solving a linear equation in the steps taken to isolate the variable. Both involve performing the same operations on both sides. The major difference is that when a linear inequality is multiplied or divided by a negative number, the inequality sign is reversed. Additionally, linear inequalities have a range of solutions instead of a single solution.

Step by step solution

01

Understand the Problems to be compared

Firstly, get a clarity about the two things you are comparing. A linear equation is an equation that makes a straight line when it's graphed. This can be solved with the aim of finding the value of the variable which makes the equation true. A linear inequality, on the other hand, is similar to a linear equation, but difference lies in the fact that instead of an equal sign, an inequality sign is used. This is solved with the aim of finding a range of values for the variable.
02

Comparing method of solving

When comparing these two, we notice that the steps for solving both linear equations and inequalities are essentially the same. Both problems involve isolating the variable to one side. This is done by performing the same operations on both sides to maintain equality or inequality.
03

Difference while dealing with multiplication or division by negative numbers

One key difference is that when a linear inequality is multiplied or divided by a negative number to solve for the variable, there is a rule that we must reverse the inequality sign. This rule does not exist when dealing with linear equations.
04

Describing the solutions

The solutions for both types of problems are also expressed differently. Linear equation has a unique solution. On the other hand, the solution to a linear inequality isn't just a definitive answer, it's a range of possible answers in the form of interval.

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Most popular questions from this chapter

Perform the indicated operations. Simplify the result, if possible. $$\left(\frac{2 x+3}{x+1} \cdot \frac{x^{2}+4 x-5}{2 x^{2}+x-3}\right)-\frac{2}{x+2}$$

If you are given a quadratic equation, how do you determine which method to use to solve it?

Will help you prepare for the material covered in the first section of the next chapter. If \(y=4-x^{2},\) find the value of \(y\) that corresponds to values of \(x\) for each integer starting with -3 and ending with 3

The rational expression $$\frac{130 x}{100-x}$$ describes the cost, in millions of dollars, to inoculate \(x\) percent of the population against a particular strain of flu. a. Evaluate the expression for \(x=40, x=80,\) and \(x=90\) Describe the meaning of each evaluation in terms of percentage inoculated and cost. b. For what value of \(x\) is the expression undefined? c. What happens to the cost as \(x\) approaches \(100 \% ?\) How can you interpret this observation?

Exercises \(142-144\) will help you prepare for the material covered in the next section. Simplify and express the answer in descending powers of \(x\) : $$ 2 x\left(x^{2}+4 x+5\right)+3\left(x^{2}+4 x+5\right) $$
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