/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 409 For the following exercises, sol... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 409

For the following exercises, solve each inequality and write the solution in interval notation. $$|3 x-4| \leq 8$$

Short Answer

Expert verified
The solution is \([-\frac{4}{3}, 4]\).

Step by step solution

01

Understand the Absolute Value Inequality

The inequality given is \(|3x - 4| \leq 8\). Since it is an absolute value inequality of the form \(|A| \leq B\), it can be split into two inequalities: 1. \(A \leq B\)2. \(A \geq -B\)These represent the range of values within the distance defined by \(B\).
02

Set up the Inequalities

From the absolute value inequality \(|3x - 4| \leq 8\), create the two inequalities:1. \(3x - 4 \leq 8\)2. \(3x - 4 \geq -8\)
03

Solve the First Inequality

Solve the inequality \(3x - 4 \leq 8\) by adding 4 to both sides to get:\(3x \leq 12\)Next, divide both sides by 3 to solve for \(x\):\(x \leq 4\)
04

Solve the Second Inequality

Now solve the inequality \(3x - 4 \geq -8\). Add 4 to both sides:\(3x \geq -4\)Next, divide both sides by 3:\(x \geq -\frac{4}{3}\)
05

Combine the Solutions

Combine the solutions from Steps 3 and 4. The solution to both inequalities is:\(-\frac{4}{3} \leq x \leq 4\).
06

Write the Solution in Interval Notation

The combined solution is written in interval notation as:\([-\frac{4}{3}, 4]\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Absolute Value
Absolute value explains the distance of a number from zero on the number line, always being non-negative. In mathematical terms, the absolute value of a number \(x\), represented as \(|x|\), is:
  • \(|x| = x\) if \(x\) is positive or zero, and
  • \(|x| = -x\) if \(x\) is negative.
In an inequality such as \(|3x - 4| \leq 8\), we interpret this as numbers whose values, regardless of sign, fall within a specified range, here between \(-8\) and \(8\). This concept lets us split one absolute value inequality into two separate inequalities, allowing for easier computation.
Interval Notation
Interval notation is a concise way of expressing a set of numbers lying within a specific range. For example, when we solve an inequality and find that \(-\frac{4}{3} \leq x \leq 4\), we use interval notation to write this as \([-\frac{4}{3}, 4]\).
This notation uses brackets and parentheses:
  • Square brackets \([\ ]\) indicate that the number is included in the interval (closed interval).
  • Parentheses \(( \ )\) suggest that the number is not included (open interval).
In our exercise, both \(-\frac{4}{3}\) and \(4\) are included, making this a closed interval.
Solving Inequalities
Solving inequalities involves finding all possible values of a variable that make an inequality true. An effective approach begins with understanding the type of inequality and then isolating the variable. For absolute value inequalities like \(|3x - 4| \leq 8\), we first split them into two separate inequalities:
  • \(3x - 4 \leq 8\)
  • \(3x - 4 \geq -8\)
Next, solve each inequality separately:
  • For \(3x - 4 \leq 8\), add 4 to each side, then divide by 3, yielding \(x \leq 4\).
  • For \(3x - 4 \geq -8\), also add 4 to each side, then divide by 3, resulting in \(x \geq -\frac{4}{3}\).
Finally, combine both solutions to represent the range of possible values for \(x\).
Mathematical Inequalities
Mathematical inequalities express the relationship that one quantity is larger or smaller than another. They are denoted by symbols like \(<\), \(>\), \(\leq\) (less than or equal to), and \(\geq\) (greater than or equal to).
When dealing with inequalities:
  • The basic rules of arithmetic apply, including addition, subtraction, multiplication, and division.
  • Adding or subtracting the same number on both sides doesn't change the inequality.
  • Multiplying or dividing by a positive number maintains the inequality direction.
  • However, multiplying or dividing both sides by a negative number reverses the inequality sign.
These rules help in manipulating and solving inequalities, as demonstrated in the step-by-step solution process of the inequality \(|3x - 4| \leq 8\).

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

For the following exercises, use the function values for \(f\) and \(g\) shown in Table 1.23 to evaluate each expression. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|}\hline x & {0} & {1} & {2} & {3} & {4} & {5} & {6} & {7} & {8} & {9} \\ \hline f(x) & {7} & {6} & {5} & {8} & {4} & {0} & {2} & {1} & {9} & {3} \\ \hline g(x) & {9} & {5} & {6} & {2} & {1} & {8} & {7} & {3} & {4} & {0} \\ \hline\end{array} $$ $$ f(g(8)) $$

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function \(f\). \(y=f(x)+5\)

For the following exercises, use the function values for \(f\) and \(g\) shown in Table 1.24 to evaluate the expressions. $$ \begin{array}{|c|c|c|c|c|c|c|c|}\hline x & {-3} & {-2} & {-1} & {0} & {1} & {2} & {3} \\ \hline f(x) & {11} & {9} & {7} & {5} & {3} & {1} & {-1} \\\ \hline g(x) & {-8} & {-3} & {0} & {1} & {0} & {-3} & {-8} \\\ \hline\end{array} $$ $$ (f \circ f)(3) $$

For the following exercises, use the function values for \(f\) and \(g\) shown in Table 1.23 to evaluate each expression. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|}\hline x & {0} & {1} & {2} & {3} & {4} & {5} & {6} & {7} & {8} & {9} \\ \hline f(x) & {7} & {6} & {5} & {8} & {4} & {0} & {2} & {1} & {9} & {3} \\ \hline g(x) & {9} & {5} & {6} & {2} & {1} & {8} & {7} & {3} & {4} & {0} \\ \hline\end{array} $$ $$ f(f(4)) $$

For the following exercises, use each pair of functions to find \(f(g(0))\) and \(g(f(0))\) $$ f(x)=\frac{1}{x+2}, g(x)=4 x+3 $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.