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Chapter 9: Problem 29

Solve each inequality and graph the solution set on a number line. Express the solution set in interval notation. $$-11<2 x-1 \leq-5$$

Short Answer

Expert verified
The solution is \( x \) belongs to the interval \( (-5, -2] \).

Step by step solution

01

Split the Compound Inequality

Break down the compound inequality into two separate inequalities: \n \( -11 < 2x - 1 \) and \( 2x - 1 鈮� -5 \).
02

Solve the first inequality

Solve the first inequality \( -11 < 2x - 1 \) for \( x \). First, add 1 to both sides of the inequality which gives us \( -10 < 2x \). Then divide both sides by 2 to get \( -5 < x \), or equivalently, \( x > -5 \).
03

Solve the second inequality

Next, solve the second inequality \( 2x - 1 鈮� -5 \) for \( x \). First, add 1 on both sides to get \( 2x 鈮� -4 \). Then, divide both sides by 2 to find \( x 鈮� -2 \).
04

Graph the solution on a number line

Plot the two solutions on a number line. From the previous steps, we know that \( x \) is strictly greater than -5 and less than or equal to -2. The number line would thus show an open circle at -5 and a closed circle at -2 with a line connecting these two points indicating all numbers between -5 and -2.
05

Express the solution in interval notation

The solution set from the previous step translates to the interval \( (-5, -2] \) in interval notation. This means that \( x \) is any number greater than -5 and up to and including -2.

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Key Concepts

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

Compound Inequalities
Compound inequalities involve more than one inequality in a single statement. Typically, these inequalities are connected by the words "and" or "or". In this exercise, we have an "and" compound inequality:
  • -11 < 2x - 1 鈮� -5
This statement is split into two separate inequalities:
  • -11 < 2x - 1
  • 2x - 1 鈮� -5
Both inequalities must be solved to find a solution that satisfies both conditions. This type of inequality typically restricts the solution to a range of values. In our exercise, the solution is between two endpoints on the real number line.
Interval Notation
Interval notation is a concise way of expressing a range of numbers on a number line. It uses brackets and parentheses:
  • Square brackets [ ] are used to include an endpoint.
  • Parentheses ( ) are used to exclude an endpoint.
For the given exercise, the solution set is expressed as
  • (-5, -2]
This notation implies:
  • "x" is greater than -5 (exclusively)
  • and less than or equal to -2 (inclusively).
Using interval notation simplifies communication of the solution set in mathematical contexts, making complex ranges easy to understand and utilize.
Graphing Inequalities
Graphing inequalities on a number line is a visual approach to understanding the solution of inequalities. It helps in identifying which sections of the number line contain solutions to the inequality:
  • Open circles are used to depict values that are not included in the solution set.
  • Closed circles show that values are included.
For our inequality, the graph shows an open circle at -5 and a closed circle at -2. A line is drawn between these two points, indicating that every number in between is part of the solution. This visualization helps solidify the understanding of the solution's constraints and boundaries.
Number Line Representation
A number line representation transforms the abstract solution of an inequality into a tangible form. It graphically maps where solutions to the inequality lie in relation to other numbers.
  • Start by marking critical point locations on the line (i.e., endpoints of the inequality).
  • Defined intervals are highlighted by shading or by drawing a connecting line.
In the given exercise,
  • an open circle at -5 indicates that -5 is not part of the solution,
  • a closed circle at -2 shows that -2 is included in the solution,
  • and the segment between them represents all numbers satisfying the inequality.
Using a number line helps visually confirm the solution's accuracy, connecting the solution to a physical picture when interpreting inequalities.

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Most popular questions from this chapter

On your next vacation, you will divide lodging between large resorts and small inns. Let \(x\) represent the number of nights spent in large resorts. Let \(y\) represent the number of nights spent in small inns. Write a system of inequalities that models the following conditions: You want to stay at least 5 nights. At least one night should be spent at a large resort. Large resorts average \(\$ 200\) per night and small inns average \(\$ 100\) per night. Your budget permits no more than \(\$ 700\) for lodging. b. Graph the solution set of the system of inequalities in part (a). c. Based on your graph in part (b), how many nights could you spend at a large resort and still stay within your budget?

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. Because the absolute value of any expression is never less than a negative number, I can immediately conclude that the inequality \(|2 x-5|-9<-4\) has no solution.

Exercises \(99-101\) will help you prepare for the material covered in the first section of the next chapter. If \(f(x)=\sqrt{3 x+12},\) find \(f(8)\)

The formula for converting Fahrenheit temperature, \(F,\) to Celsius temperature, \(C\), is $$C=\frac{5}{9}(F-32)$$ If Celsius temperature ranges from \(15^{\circ}\) to \(35^{\circ},\) inclusive, what is the range for the Fahrenheit temperature? Use interval notation to express this range.

In more U.S. marriages, spouses have different faiths. The bar graph shows the percentage of households with an interfaith marriage in 1988 and \(2008 .\) Also shown is the percentage of households in which a person of faith is married to someone with no religion. (GRAPH CANT COPY) The formula $$1-\frac{1}{2} x+2=$$ models the percentage of U.S. households with an interfaith marriage, \(I, x\) years after \(1988 .\) The formula $$N=\frac{1}{4} x+6$$ models the percentage of U.S. households in which a person of faith is married to someone with no religion, \(N, x\) years after I988. Use these models to solve. a. In which years will more than \(34 \%\) of U.S. households have an interfaith marriage? b. In which years will more than \(15 \%\) of U.S. households have a person of faith married to someone with no religion? c. Based on your answers to parts (a) and (b), in which years will more than \(34 \%\) of households have an interfaith marriage and more than \(15 \%\) have a faith/no religion marriage? d. Based on your answers to parts (a) and (b), in which years will more than \(34 \%\) of households have an interfaith marriage or more than \(15 \%\) have a faith/no religion marriage?
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