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Chapter 5: Problem 40

Graph the solution set. \(|x|+1>3\)

Short Answer

Expert verified
The solution set is \(x < -2\) or \(x > 2\).

Step by step solution

01

Understand the Absolute Value

The expression \(|x|\) represents the absolute value of \(x\). This means that \(|x|\) is always non-negative regardless of whether \(x\) is positive or negative.
02

Set Up Inequality

The given inequality is \(|x| + 1 > 3\). To isolate the absolute value, subtract 1 from both sides: \(|x| > 2\).
03

Split the Absolute Value Inequality

The inequality \(|x| > 2\) can be split into two separate inequalities: \(x > 2\) and \(x < -2\). This is because the absolute value inequality represents a distance from zero.
04

Graph the Solution

On a number line, graph the solution of the inequalities \(x > 2\) and \(x < -2\). These are both open intervals. The graph will include all numbers to the right of 2 and all numbers to the left of -2, not including the points 2 and -2 themselves.

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Key Concepts

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

Graphing Inequalities
Graphing inequalities allows us to visually represent the solutions to mathematical inequalities. We can see where the values fall on a number line or coordinate plane. When graphing inequalities, it's crucial to understand the difference between open and closed intervals:
  • Open intervals do not include the boundary points. They are represented with open circles.

  • Closed intervals include the boundary points, marked with filled circles.

For example, the inequality \(|x| > 2\) gives us open intervals at 2 and -2. This is shown with open circles at these points, and shading the areas that are greater than 2 and less than -2.
Graphing inequalities is a powerful tool that makes it easier to see solutions and understand inequality relationships.
Absolute Value
The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always non-negative. For instance, \(|3| = 3\) and \(|-3| = 3\).

For an inequality, \(|x| > 2\), we consider all positions on the number line where the distance from zero is more than 2. This splits the inequality into:
  • \ (x > 2) \

  • \ (x < -2) \

Absolute values can seem tricky because they deal with both positive and negative solutions. But they simplify to a straightforward principle 鈥� how far a number is from zero.
Number Line
A number line is a visual representation of numbers arranged in order from left to right. The center or reference point of a number line is zero. To graph inequalities on a number line:
  • Identify the key points needed (such as -2 and 2 for \(|x| > 2\))

  • Determine if those points are included (closed circles) or excluded (open circles)

  • Shade regions that satisfy the inequality

For instance, for the inequality \(|x| > 2\), place open circles at -2 and 2 and shade all numbers to the left of -2 and to the right of 2. This shading visually represents all solutions where the inequality holds.
Using a number line helps to clarify and solve inequalities more effectively.

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Most popular questions from this chapter

A furniture manufacturer builds tables. The cost for materials and labor to build a kitchen table is \(\$ 240\) and the profit is \(\$ 160 .\) The cost to build a dining room table is \(\$ 320\) and the profit is \(\$ 240\). (See Examples \(2-3)\) Let \(x\) represent the number of kitchen tables produced per month. Let \(y\) represent the number of dining room tables produced per month. a. Write an objective function representing the monthly profit for producing and selling \(x\) kitchen tables and \(y\) dining room tables. b. The manufacturing process is subject to the following constraints. Write a system of inequalities representing the constraints. \- The number of each type of table cannot be negative. \- Due to labor and equipment restrictions, the company can build at most 120 kitchen tables. \- The company can build at most 90 dining room tables. \- The company does not want to exceed a monthly cost of \(\$ 48,000\). c. Graph the system of inequalities represented by the constraints. d. Find the vertices of the feasible region. e. Test the objective function at each vertex. f. How many kitchen tables and how many dining room tables should be produced to maximize profit? (Assume that all tables produced will be sold.) g. What is the maximum profit?

Jonas performed an experiment for his science fair project. He learned that rinsing lettuce in vinegar kills more bacteria than rinsing with water or with a popular commercial product. As a follow-up to his project, he wants to determine the percentage of bacteria killed by rinsing with a diluted solution of vinegar. a. How much water and how much vinegar should be mixed to produce 10 cups of a mixture that is \(40 \%\) vinegar? b. How much pure vinegar and how much \(40 \%\) vinegar solution should be mixed to produce 10 cups of a mixture that is \(60 \%\) vinegar?

Solve the system of equations by using the addition method. (See Examples \(3-4)\) $$ \begin{array}{l} 3 x-7 y=1 \\ 6 x+5 y=-17 \end{array} $$

A farmer has 1200 acres of land and plans to plant corn and soybeans. The input cost (cost of seed, fertilizer, herbicide, and insecticide) for 1 acre for each crop is given in the table along with the cost of machinery and labor. The profit for 1 acre of each crop is given in the last column. $$ \begin{array}{|l|c|c|c|} \hline & \begin{array}{c} \text { Input Cost } \\ \text { per Acre } \end{array} & \begin{array}{c} \text { Labor/Machinery } \\ \text { Cost per Acre } \end{array} & \begin{array}{c} \text { Profit } \\ \text { per Acre } \end{array} \\ \hline \text { Corn } & \$ 180 & \$ 80 & \$ 120 \\ \hline \text { Soybeans } & \$ 120 & \$ 100 & \$ 100 \\ \hline \end{array} $$ Suppose the farmer has budgeted a maximum of $$\$ 198,000$$ for input costs and a maximum of $$\$ 110,000$$ for labor and machinery. a. Determine the number of acres of each crop that the farmer should plant to maximize profit. (Assume that all crops will be sold.) b. What is the maximum profit? c. If the profit per acre were reversed between the two crops (that is, $$\$ 100$$ per acre for corn and $$\$ 120$$ per acre for soybeans), how many acres of each crop should be planted to maximize profit?

Juan borrows \(\$ 100,000\) to pay for medical school. He borrows part of the money from the school whereby he will pay \(4.5 \%\) simple interest. He borrows the rest of the money through a government loan that will charge him \(6 \%\) interest. In both cases, he is not required to pay off the principal or interest during his 4 yr of medical school. However, at the end of \(4 \mathrm{yr}\), he will owe a total of \(\$ 19,200\) for the interest from both loans. How much did he borrow from each source?
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