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Chapter 9: Problem 17

Graph each inequality. $$x \leq 2$$

Short Answer

Expert verified
The graph should include a solid line at \(x = 2\) and the area to the left of and including the line should be shaded to represent the solution to the inequality \(x \leq 2\).

Step by step solution

01

Draw the number line

Start by drawing the number line. Mark the point '2' as this is significant in the inequality.
02

Draw the line

Draw a vertical line at \(x = 2\) on the graph.
03

Decide on line type

The line must be solid because the inequality is '\(x \leq 2\)', not '\(x < 2\)'. The 'less than or equal to' symbol suggests that 2 is included in the solution, thus a solid line is used.
04

Determine solution area

The solution to the inequality \(x \leq 2\) includes all values of \(x\) that are less than 2. Therefore, shade the area to the left of and including the line \(x=2\).

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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 a simple visual representation used to show numerical values in a straight, horizontal line. It helps in visualizing inequalities and understanding their solutions quickly. To represent numbers, we place them at evenly spaced intervals along the line. The point of interest in our example is the number 2. When dealing with inequalities, like in this task, we use the number line to clearly mark the significant points where the inequality changes, allowing us to efficiently interpret the range of solutions. Make sure your number line covers a reasonable range to include all relevant values.
Solid Line
When graphing inequalities on a number line, the type of line we use is crucial. For the inequality given, \(x \leq 2\), a **solid line** indicates that the number itself is included in the solutions. A solid line at a certain value, like at \(x=2\), illustrates that the inequality allows for "equal" as part of "less than or equal to". So, every point on the line itself is a valid solution to the inequality. If the inequality were strict, such as \(x < 2\), you would use a hollow or dashed line instead to show that the point isn't included.
Solution Area
The solution area of an inequality is the part of the number line that includes all values that satisfy the inequality. For the inequality \(x \leq 2\), the solution area is all the numbers to the left of and including 2. This shows that x can be 2 or any number smaller than 2. By identifying the solution area, you're effectively identifying the range of all possible solutions that make the inequality true. It's helpful to shade this section to clearly distinguish which parts of the number line are part of the solution, enhancing clarity in your graph.
Shading
Shading on a number line or a graph is an effective method to visually indicate the solution set of an inequality. Once you've drawn your solid line at the point of interest, it's time to shade. For \(x \leq 2\), you shade to the left of the solid line, including the line itself. This shading helps to visually communicate where all the potential solutions lie. It acts as a quick reference to understand which values satisfy the inequality without having to analyze the numbers individually. Effective shading eliminates confusion and makes your graph easier to interpret for anyone reviewing your work.

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Most popular questions from this chapter

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I'll win the contest if I can complete the crossword puzzle in 20 minutes plus or minus 5 minutes, so my winning time, \(x,\) is modeled by \(|x-20| \leq 5\)

Graphing utilities can be used to shade regions in the rectangular coordinate system, thereby graphing an inequality in two variables. Read the section of the user's mamual for your graphing utility that describes how to shade a region. Then use your graphing unility to graph the inequalities. $$3 x-2 y \geq 6$$

Graphing utilities can be used to shade regions in the rectangular coordinate system, thereby graphing an inequality in two variables. Read the section of the user's mamual for your graphing utility that describes how to shade a region. Then use your graphing unility to graph the inequalities. $$2 x+y \leq 6$$

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