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Chapter 9: Problem 23

Solve each compound inequality. Use graphs to show the solution set to each of the two given inequalities, as well as a third graph that shows the solution set of the compound inequality. Except for the empty set, express the solution set in interval notation. $$x-1 \leq 7 x-1 \text { and } 4 x-7<3-x$$

Short Answer

Expert verified
The solution of the compound inequality is \](-\infty, 2)\[.

Step by step solution

01

Solve each inequality

We begin by separating the compound inequalities into two inequalities and solving them individually. For \(x-1 \leq 7\), we add 1 to both sides to isolate \(x\):\[x \leq 8\]For \(4x-7<3-x\), we add \(x\) and 7 to both sides:\[5x < 10 \rightarrow x < 2\]
02

Graph the inequality solutions

Now, we graph the solutions of the two inequalities. For \(x \leq 8\), the graph is a number line with a closed circle at 8 and shading to the left, representing all \(x\) values less than or equal to 8. For \(x < 2\), the graph is a number line with an open circle at 2 and shading to the left, representing all \(x\) values strictly less than 2.
03

Graph the solution to the compound inequality

Since the original problem is a compound inequality with 'and', we take the intersection of the two individual solution sets. This means we look at where the shading overlaps in our separate graphs. The overlap occurs for \(x\) values that are less than 2.
04

Write the solution in interval notation

Finally, we express our solution set in interval notation. For \(x < 2\), this is \](-\infty, 2)\[. The symbol \](-\infty, 2)\[ represents all numbers less than 2.

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Key Concepts

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

Inequality Solution
Solving compound inequalities involves handling two or more simple inequalities connected by the words "and" or "or." In this context, "and" indicates that the solution must satisfy both inequalities simultaneously. For instance, given the compound inequality \(x - 1 \leq 7\) and \(4x - 7 < 3 - x\), we solve each inequality separately.
  • For \(x - 1 \leq 7\): Add 1 to both sides to isolate \(x\), resulting in \(x \leq 8\).
  • For \(4x - 7 < 3 - x\): Add \(x\) and 7 to both sides, simplifying to \(5x < 10\), which further reduces to \(x < 2\) after division by 5.
These individual steps highlight the process of finding where the variable \(x\) must lie to make both original inequalities true. Solving each part is crucial to understanding where overlaps or intersections, in the case of "and," occur in the solution space.
Graphing Inequalities
Graphing inequalities visually represents the solutions on a number line, making it easier to discern overlapping solution sets. Each inequality displays a specific segment of the number line:
  • For \(x \leq 8\), the graph involves a closed circle at 8, showing that 8 is included in the solution set, and shading to the left towards negative infinity.
  • For \(x < 2\), the graph has an open circle at 2, indicating that 2 is not part of the solution set, with shading extending to the left again.
To find the solution of a compound inequality linked by "and," we look for where these shaded regions overlap. In this example, the overlap happens at all points less than 2. Graphing not only confirms our algebraic solution but also provides a visual method to check the consistency of our results.
Interval Notation
Interval notation is a concise way to represent ranges of solutions for inequalities. It uses parentheses \(()\) and brackets \([]\) to describe open and closed endpoints, respectively. For compound inequalities, once the solution range is determined, translating this to interval notation simplifies the expression of the solution set:
  • For the solution \(x < 2\), the interval notation is \((-\infty, 2)\). Here, the parenthesis instead of a bracket indicates that 2 is not part of the solution set, matching the open circle on the graph.
Interval notation is powerful not just for its brevity but for its clear communication of which endpoint values are included (closed with brackets) or excluded (open with parentheses) in the set. This format is essential for efficiently conveying the results of solved inequalities.

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Values of \(-5\) and 5 satisfy \(|x|=5,|x| \leq 5,\) and \(|x| \geq-5\)

Write the given sentences as a system of linear inequalities in no variables. Then graph the system. The sum of the \(x\) -variable and the \(y\) -variable is at most 4 The \(y\) -variable added to the product of 3 and the \(x\) -variable does not exceed 6

The graphs of solution sets of systems of inequalities involve finding the intersection of the solution sets of nwo or more incqualities. By contrast, in Exercises \(53-54\) you will be graphing the union of the solution sets of two inequalities. \(\left\\{\begin{array}{l}6 x-y \leq 24 \\ 6 x-y \geq 24\end{array}\right.\)

On the first four exams, your grades are \(82,75,80,\) and 90. There is still a final exam, and it counts as two grades. You are hoping to earn a \(\mathrm{B}\) in the course: This will occur if the average of your six exam grades is greater than or equal to 80 and less than \(90 .\) What range of grades on the final exam will result in earning a B? 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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