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Chapter 1: Problem 36

Graph the solution set of each inequality on the real number line. $$-4 \leq x \leq-1$$

Short Answer

Expert verified
The graph of the solution set for the inequality \(-4 \leq x \leq -1\) is a real number line with points -4 and -1 marked, both using filled circles to represent inclusive limits, and the segment between the two points is highlighted, representing all valid values of x.

Step by step solution

01

Identify the Critical Values

The critical values in this inequality are -4 and -1. These are the values that define the lower and upper limit of the interval for the variable x.
02

Draw the Number Line

Draw a number line with sufficient range to accommodate for all critical values. -4 and -1 should fall within your drawn number line.
03

Plot the Critical Values

Mark points for -4 and -1 on the number line. Since the equal sign (鈮�) is used in the inequality, represent these points with a filled circle - \([ \) or \( ] \)\.
04

Highlight the Interval

The interval between -4 and -1 on the number line represents all possible values for x in the inequality. Highlight this interval to visualize the solution to the inequality.

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Key Concepts

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

Real Number Line
The real number line is a visual representation of all real numbers laid out in a straight line. This line is used to illustrate inequalities and can help us understand the set of possible values a variable can take. When drawing a real number line:
  • Include both positive and negative numbers, as well as zero.
  • Place each number equally spaced along the line for clarity.
  • Use arrows on both ends to indicate that the real numbers extend infinitely in both directions.
In the exercise given, the numbers -4 and -1 are the critical points that we need to place on our real number line. By marking these points, we clearly define the segment of the line that corresponds to the solution of the inequality. You can also think of the number line as a ruler, where each point or tick represents a specific numerical value.
Interval Notation
Interval notation is a method of writing subsets of the real number line. It's a concise way to describe the set of solutions that satisfy an inequality. For the inequality \[-4 \leq x \leq-1\], the interval notation is written as \([-4, -1]\). This interval notation means:
  • -4 is included in the set, as indicated by the square bracket.
  • -1 is also included, due to the other square bracket.
Square brackets, \([\) and \([)\), denote that the endpoints are included in the interval. When endpoints are not included, parentheses, \(\((\) or \(\))\), are used instead. In this case, since the inequality includes the equal sign, both endpoints -4 and -1 are part of the solution, thus requiring brackets in interval notation.
Critical Values
Critical values are specific numbers that play a key role in defining intervals or regions on the number line for inequalities. In our inequality \(-4 \leq x \leq -1\), the critical values are -4 and -1. These numbers define the limit or boundary of the range of solutions:
  • -4 is the smallest value in the range that's allowed by the inequality.
  • -1 is the largest value in the range that's allowed by the inequality.
Once we determine the critical values, we can illustrate the solution set on the number line by marking these values and shading between them. Recognizing critical values is crucial because they provide the anchor points between which the inequality holds true. Ensuring they are marked correctly is vital for accurately depicting the solution of the inequality.

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Most popular questions from this chapter

Solve the inequality. Express your answer in interval notation. $$\frac{2}{3} x+3 \leq 5$$

The following function gives the cost for printing \(x\) copies of a 120 -page book. The minimum number of copies printed is 200 , and the maximum number printed is 1000 $$f(x)=\left\\{\begin{array}{ll}5 x, & \text { if } 200 \leq x<500 \\\4.5 x, & \text { if } 500 \leq x<750 \\\4 x, & \text { if } 750 \leq x \leq 1000\end{array}\right.$$ (a) Find the cost for printing 400 copies of the book. (b) Find the cost for printing 620 copies of the book. (c) Which is more expensive-printing 700 copies of the book or printing 750 copies of the book?

Graph the piecewise-defined function using a graphing utility. The display should be in DOT mode. $$f(x)=\left\\{\begin{array}{ll} x^{2}, & \text { if }-2 \leq x<0 \\ -x+1, & \text { if } 0 \leq x<2.5 \\ x-3.5, & \text { if } x \geq 2.5 \end{array}\right.$$

Solve the inequality. Express your answer in interval notation, and graph the solution set on the number line. $$|2-3 x| \leq 10$$

Graph the function by hand. $$f(x)=\left\\{\begin{array}{ll} -x, & -3 \leq x<3 \\ 2 x, & x \geq 3 \end{array}\right.$$
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