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Chapter 9: Problem 45

Solve and graph the solution set on a number line. $$|2 x-6|<8$$

Short Answer

Expert verified
The solution is \( -1 < x < 7 \), which can be represented on a number line as a line segment between points -1 and 7 (exclusive)

Step by step solution

01

Simplify the inequalities

Remove the absolute value sign from \(|2x - 6| < 8 \). Thus, the problem becomes \(-8 < 2x - 6 < 8\)
02

Solve for x

Add 6 to all three sides of the inequality to isolate the variable, x, resulting in \(-8 + 6 < 2x < 8 + 6\) which simplifies to \(-2 < 2x < 14 \). Then, divide all three sides by 2 to generate the final inequality \(-1 < x < 7\)
03

Graph the solution on a number line

Draw a line to represent the number line. Mark the points -1 and 7 as open circles (since the inequality does not include these values). Fill in the line segment between -1 and 7 to denote all the numbers between -1 and 7 (exclusive) are part of the solution set

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Key Concepts

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

Graphing Inequalities
Graphing inequalities is a visual way to understand the range of solutions for an inequality. When dealing with absolute value inequalities like \(|2x - 6| < 8\), the task is to express which values of \(x\) make the inequality true. Graphing helps illustrate this span on a number line, making it easier to see.
  • Begin with solving the inequality step by step to isolate the variable.
  • Once solved, express the solution in inequality form.
  • Transfer this solution to the number line graphically.
When graphing, represent the endpoints of the range as open or closed circles, depending on the inequality sign used鈥攐pen for \(<\) and closed for \(\leq, \geq\).
In our example, \(-1 < x < 7\), the points \(-1\) and \(7\) are marked as open circles. The line between them is shaded to represent every value of \(x\) within that span fulfilling the inequality.
Solving Inequalities
Solving inequalities involves finding all values of the variable that make the inequality true. Much like solving an equation, but with a crucial difference when you multiply or divide by a negative number. Let's walk through the process using our example \(|2x - 6| < 8\):
  • Start by removing the absolute value, breaking it into two inequalities: \(-8 < 2x - 6 < 8\).
  • Simplify step by step. Here, you can add \(6\) to eliminate the constant term from both inequalities.
  • Divide by \(2\) to solve for \(x\). Remember to treat each segment of the inequality independently.
  • The solution is \(-1 < x < 7\).
Remember, when you divide or multiply by a negative number, invert the inequality signs. This caution ensures the solution set remains accurate.
Number Line Representation
A number line offers a clear visual method for representing the solution set of inequalities. After solving an inequality, graphing it on a number line can make the solution intuitive and accessible. To do this with our solution set \(-1 < x < 7\):
  • Draw a horizontal line and mark critical points like \(-1\) and \(7\).
  • Use open circles to indicate these points are not included in the solution set (as \(x\) is strictly greater than \(-1\) and less than \(7\)).
  • Shade the portion of the line between these points. This shaded section depicts every number that satisfies the inequality.
This visual representation aids understanding by highlighting all numbers fulfilling the condition set by the inequality. It's a straightforward way to comprehend the breadth of potential solutions at a glance.

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Most popular questions from this chapter

Simplify: \(4-[2(x-4)-5] .\) (Section 1.8, Example 11)

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 \(33 \%\) of U.S. households have an interfaith marriage? b. In which years will more than \(14 \%\) 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 \(33 \%\) of households have an interfaith marriage and more than \(14 \%\) have a faith/no religion marriage? d. Based on your answers to parts (a) and (b), in which years will more than \(33 \%\) of households have an interfaith marriage or more than \(14 \%\) have a faith/no religion marriage?

Explain how to graph \(x-2 y<4\)

Explain how to solve \(1<2 x+3<9\)

Will help you prepare for the material covered in the next section. Find all values of \(x\) satisfying \(3 x-1=x+5\) or \(3 x-1=-(x+5)\)
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