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Chapter 1: Problem 83

Solve each equation or inequality. $$ |2 x+1|+3>8 $$

Short Answer

Expert verified
x > 2 or x < -3

Step by step solution

01

Isolate the Absolute Value Expression

Subtract 3 from both sides of the inequality: |2x + 1| + 3 - 3 > 8 - 3 This simplifies to: |2x + 1| > 5
02

Set Up Two Inequalities

The definition of absolute value tells us that |2x + 1| > 5 means two different inequalities: 2x + 1 > 5 and 2x + 1 < -5
03

Solve the First Inequality

Solve the first inequality, 2x + 1 > 5: Subtract 1 from both sides:2x > 4 Divide both sides by 2:x > 2
04

Solve the Second Inequality

Solve the second inequality, 2x + 1 < -5: Subtract 1 from both sides:2x < -6 Divide both sides by 2:x < -3
05

Combine the Solutions

Combine the solutions from the two inequalities: x > 2 or x < -3

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Key Concepts

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

absolute value inequalities
Absolute value inequalities can seem tricky at first, but they are manageable if approached step-by-step. The absolute value of a number is its distance from zero on the number line, regardless of direction. This means \(|x| = a\) implies that 'x' could be either 'a' or '-a'.

For example, \(|2x + 1| > 5\) signifies that \((2x + 1)\) can be more than 5 in a positive direction, or less than -5 in the negative direction. In essence, these inequalities split into two simpler expressions that can be solved separately.
isolation of absolute value expressions
The first step in solving absolute value inequalities is to isolate the absolute value expression on one side of the inequality. This is similar to isolating a variable in a simple equation.

In the given problem, we start with: \(|2x + 1| + 3 > 8\).

To isolate \(|2x + 1|\), subtract 3 from both sides: \(|2x + 1| + 3 - 3 > 8 - 3\), which simplifies to \(|2x + 1| > 5\).

Isolating the absolute value makes it easier to split this into two separate inequalities as per the definition of absolute value.
combining inequalities
After isolating the absolute value, we need to set up and solve two different inequalities.

From \(|2x + 1| > 5\), we derive: \(2x + 1 > 5\) and \(2x + 1 < -5\).

Let's solve each one step-by-step:
  • For \(2x + 1 > 5\):
    • Subtract 1 from both sides: \(2x > 4\)
    • Divide by 2: \(x > 2\)
  • For \(2x + 1 < -5\):
    • Subtract 1 from both sides: \(2x < -6\)
    • Divide by 2: \(x < -3\)


Finally, we combine these solutions to express all possible

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Most popular questions from this chapter

Find the open interval in which \(x\) must lie in order for the given condition to hold. \(y=4 x-8,\) and the difference of \(y\) and 3 is less than 0.001.

The 10 tallest buildings in Houston, Texas, are listed along with their heights. $$ \begin{array}{|l|c|} \hline \quad {\text { Building }} & \text { Height (in feet) } \\ \hline \text { JPMorgan Chase Tower } & 1002 \\ \text { Wells Fargo Plaza } & 992 \\ \text { Williams Tower } & 901 \\ \text { Bank of America Center } & 780 \\ \text { Texaco Heritage Plaza } & 762 \\ \text { 609 Main at Texas } & 757 \\ \text { Enterprise Plaza } & 756 \\ \text { Centerpoint Energy Plaza } & 741 \\ \text { 1600 Smith St. } & 732 \\ \text { Fulbright Tower } & 725 \\ \hline \end{array} $$ Use this information. Work each of the following. (a) Write an absolute value inequality that describes the height of a building that is not within \(95 \mathrm{ft}\) of the average. (b) Solve the inequality from part (a). (c) Use the result of part (b) to list the buildings that are not within \(95 \mathrm{ft}\) of the average. (d) Confirm that the answer to part (c) makes sense by comparing it with the answer to Exercise 131 .

Solve each problem. How many liters of a \(10 \%\) alcohol solution must be mixed with \(40 \mathrm{~L}\) of a \(50 \%\) solution to obtain a \(40 \%\) solution?

Solve the linear equation. Graph the solution set on a number line. $$ 5(x+3)-2(x-4)=2(x+7) $$

Find the open interval in which \(x\) must lie in order for the given condition to hold. \(y=2 x+1,\) and the difference of \(y\) and 1 is less than 0.1 .
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