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Chapter 3: Problem 5

Find the graphical solution of each inequality. $$ x \leq-3 $$

Short Answer

Expert verified
The graphical solution of the inequality \(x \leq -3\) is a number line with a solid circle at -3 and shading to the left of -3, representing all the x-values that are less than or equal to -3.

Step by step solution

01

Set up a number line

Draw a horizontal straight line, and mark the point -3 on the number line as our point of interest. Include a few more points on the left and right for reference (e.g., -4, -2).
02

Illustrate the inequality

Since the inequality requires x values to be less than or equal to -3, we will show this on the number line by shading the region to the left of -3. That region represents all the x-values that are less than -3.
03

Specify the end-point

The inequality includes the equal sign (\(\leq\)), so it means that -3 is also a solution for the inequality. To indicate this, we will use a solid circle (or dot) on the number line at the point -3.
04

Final graphical solution

The final graphical solution is a number line with a solid circle at -3 and shading to the left of -3, representing the x-values in the inequality \(x\leq -3\).

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Key Concepts

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

Number Line
A number line is one of the fundamental tools used in mathematics to represent numbers visually. Think of it as a line that extends infinitely in both the positive and negative directions, with a scale that usually counts up by ones. To use a number line for solving inequalities, you first need to draw this line and then accurately mark and label significant numbers related to the problem at hand.

For instance, if you're working with the inequality \( x \leq -3 \), like in our exercise, you would draw your line horizontally and then put a mark at -3. It's good practice to also mark a few numbers to the left and to the right to give you a sense of placement鈥攆or example, -4, -2, and so on. A correctly drawn number line as a starting point is essential for clearly visualizing and solving inequalities.
Inequality Representation
Representing inequalities on a number line is a visual approach to understanding which numbers satisfy the given inequality. An inequality like \(x \leq -3\) informs us that the values for \(x\) include -3 and any number less than -3.

To represent this graphically, you can imagine each point on the line as a potential solution. Since the inequality includes -3 (as indicated by the \(less than or equal to\) sign), we represent this with a solid circle at -3. Then, we shade or color in the entire line to the left of -3 to show all possible numbers that are included in the inequality. This shaded line indicates the range of numbers that satisfy the inequality, ensuring that students can clearly see the set of all acceptable solutions.
Linear Inequalities
Linear inequalities are algebraic expressions that involve a linear function and a comparison operator, such as <, >, \(\leq\), or \(\geq\). Unlike linear equations, which have a single solution, linear inequalities like the one we're considering, \(x \leq -3\), have a range of solutions.

Students must understand that the solution to a linear inequality is not just a number but a range of numbers that satisfy the inequality. The concepts of 'less than' or 'greater than' and inclusive or exclusive (whether to include the boundary point) are critical when graphing these solutions. The main goal when dealing with linear inequalities is to isolate the variable on one side of the inequality, much like solving a regular equation, but keeping in mind the comparison operator and what it implies for the range of solutions.

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Most popular questions from this chapter

MANUFACTURING-SHIPPING CosTS Steinwelt Piano manufactures uprights and consoles in two plants, plant I and plant II. The output of plant \(I\) is at most \(300 /\) month, whereas the output of plant II is at most \(250 /\) month. These pianos are shipped to three warehouses that serve as distribution centers for the company. To fill current and projected future orders, warehouse A requires a minimum of 200 pianos/month, warehouse B requires at least 150 pianos/month, and warehouse \(\mathrm{C}\) requires at least 200 pianos/month. The shipping cost of each piano from plant I to warehouse A, warehouse \(\mathrm{B}\), and warehouse \(\mathrm{C}\) is \(\$ 60, \$ 60\), and \(\$ 80\), respectively, and the shipping cost of each piano from plant II to warehouse A, warehouse \(\mathrm{B}\), and warehouse \(\mathrm{C}\) is \(\$ 80, \$ 70\), and \(\$ 50\), respectively. What shipping schedule will enable Steinwelt to meet the warehouses' requirements while keeping shipping costs to a minimum?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The problem Maximize \(\quad P=x y\) $$ \text { subject to } \begin{aligned} & 2 x+3 y \leq 12 \\ & 2 x+y \leq 8 \\ & x \geq 0, y \geq 0 \end{aligned} $$ is a linear programming problem.

Solve each linear programming problem by the method of corners. $$ \begin{array}{lr} \text { Maximize } P=x+3 y \\ \text { subject to } \quad 2 x+y \leq 6 \\ x+y \leq 4 \\ x \leq 1 \\ x \geq 0, y \geq 0 \end{array} $$

You are given a linear programming problem. a. Use the method of corners to solve the problem. b. Find the range of values that the coefficient of \(x\) can assume without changing the optimal solution. c. Find the range of values that resource 1 (requirement 1) can assume. d. Find the shadow price for resource 1 (requirement 1). e. Identify the binding and nonbinding constraints. $$ \begin{array}{ll} \text { Maximize } & P=3 x+4 y \\ \text { subject to } & 2 x+3 y \leq 12 \\ & 2 x+y \leq 8 \\ & x \geq 0, y \geq 0 \end{array} $$

INVESTMENTS-AssET AuLocation Ashley has earmarked at most \(\$ 250,000\) for investment in three mutual funds: a money market fund, an international equity fund, and a growth-and-income fund. The money market fund has a rate of return of \(6 \% / y e a r\), the international equity fund has a rate of return of \(10 \% /\) year, and the growth-and-income fund has a rate of return of \(15 \% /\) year. Ashley has stipulated that no more than \(25 \%\) of her total portfolio should be in the growth-and-income fund and that no more than \(50 \%\) of her total portfolio should be in the international equity fund. To maximize the return on her investment, how much should Ashley invest in each type of fund?
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