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Chapter 8: Problem 75

Solve and graph. $$ |3 a-4|+2 \geq 8 $$

Short Answer

Expert verified
The solution is \( a \leq -\frac{2}{3} \ or \ a \geq \frac{10}{3} \).

Step by step solution

01

Isolate the Absolute Value Expression

Subtract 2 from both sides of the inequality to isolate the absolute value term. This gives: \[ |3a - 4| \geq 6 \]
02

Split the Inequality into Two Cases

Since the absolute value expression can be either positive or negative, consider two cases for 3a - 4:Case 1: \[ 3a - 4 \geq 6 \]Case 2: \[ 3a - 4 \leq -6 \]
03

Solve Case 1

For the first case, add 4 to both sides, then divide by 3:\[ 3a - 4 \geq 6 \ 3a \geq 10 \ a \geq \frac{10}{3} \]
04

Solve Case 2

For the second case, add 4 to both sides, then divide by 3:\[ 3a - 4 \leq -6 \ 3a \leq -2 \ a \leq \- \frac{2}{3} \]
05

Combine the Solutions

Combine the two solutions to form the final compound inequality:\[ a \leq - \frac{2}{3} \ or \ a \geq \frac{10}{3} \]
06

Graph the Solution

Graph the inequality on a number line. Indicate that the solution is all values of \( a \) less than or equal to \( -\frac{2}{3} \) and all values of \( a \) greater than or equal to \( \frac{10}{3} \). Use closed circles at \( -\frac{2}{3} \) and \( \frac{10}{3} \).

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Key Concepts

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

inequality
Inequalities are mathematical statements that show the relationship between two values when they are not equal. They are written using symbols like < (less than), > (greater than), 鈮� (less than or equal to), and 鈮� (greater than or equal to).
absolute value
Absolute value represents the distance of a number from zero on a number line, regardless of direction. It is always non-negative. For any real number x, the absolute value of x is denoted as |x|.
graphing inequalities
Graphing inequalities on a number line helps to visualize the range of solutions. For example, when graphing a solution like \( a \leq - \frac{2}{3} \ or\ a\ \geq \frac{10}{3} \), we use closed circles on \( - \frac{2}{3} \) and \( \frac{10}{3} \), and shade the portions of the number line where the inequality holds.
algebraic expressions
Algebraic expressions comprise variables, constants, and arithmetic operations. Solving absolute value inequalities often involves manipulating these expressions to isolate the absolute value term, then solving the resulting linear inequalities.

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Most popular questions from this chapter

Use a graphing calculator to graph each function and find solutions of \(f(x)=0 .\) Then solve the inequalities \(f(x)<0\) and \(f(x)>0\). $$ f(x)=x-\sqrt{x}, x \geq 0 $$

Solve. $$ 7.34 x^{2}-16.55 x-3.89 \geq 0 $$

Solve. $$ \frac{1}{3} x^{3}-x+\frac{2}{3}>0 $$

Solve. $$ \left|\frac{2 x-1}{3}\right|=4 $$

For \(f(x)\) as given, use interval notation to write the domain of \(f\) $$ f(x)=\frac{2}{x+3} $$
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