/*! 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 38 Solve each inequality. $$3 t-8... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 38

Solve each inequality. $$3 t-8 \leq t-7$$

Short Answer

Expert verified
t \leq \frac{1}{2}

Step by step solution

01

Isolate variable term

Subtract \( t \) from both sides of the inequality to begin isolating the variable term. \[ 3t - 8 - t \leq t - 7 - t \]\ which simplifies to: \[ 2t - 8 \leq -7 \]
02

Add constant to both sides

Add 8 to both sides of the inequality to remove the constant from the left side: \[ 2t - 8 + 8 \leq -7 + 8 \] which simplifies to: \[ 2t \leq 1 \]
03

Solve for the variable

Divide both sides of the inequality by 2 to solve for \( t \): \[ \frac{2t}{2} \leq \frac{1}{2} \]\ which simplifies to: \[ t \leq \frac{1}{2} \]

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.

algebraic inequalities
Algebraic inequalities are mathematical statements that relate expressions using inequality signs like \(\leq\), \(\geq\), \(<\), and \(>\). They help us understand the range of values that variables can take instead of a single fixed value. In the exercise, we worked on an inequality where we needed to find all possible values of \(t\) that satisfy the expression \(3t - 8 \leq t - 7\).
isolation of variable
Isolating the variable is a key step in solving equations and inequalities. The goal is to get the variable by itself on one side of the inequality sign. In our example, we started by subtracting \(t\) from both sides:
\[ 3t - 8 - t \leq t - 7 - t \]
This simplifies to:
\[ 2t - 8 \leq -7 \]
By carefully performing operations on both sides, we ensure the inequality remains balanced.
basic algebra
Basic algebra involves solving equations and inequalities by performing operations like addition, subtraction, multiplication, and division. After isolating the variable, we added 8 to both sides:
\[ 2t - 8 + 8 \leq -7 + 8 \]
which simplifies to:
\[ 2t \leq 1 \]
The final step was to divide by 2:
\[ \frac{2t}{2} \leq \frac{1}{2} \]
giving us
\[ t \leq \frac{1}{2} \]
This thorough approach keeps equations and inequalities manageable and solutions accurate.

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

Solve each of the problems algebraically. That is, set up an equation and solve it. Be sure to clearly label what the variable represents. Round your answer to the nearest tenth where necessary. A business buys a piece of machinery for 32,800 dollar. For tax purposes the business depreciates the value \(V\) of the machine after \(y\) years according to the formula \(V=32,800-1,850 y .\) After how many years is the machine worth 20,000 dollar?

A truck is loaded with boxes of two different weights. Some of the boxes weigh \(6.58 \mathrm{kg}\) each, and the rest of the boxes weigh \(9.32 \mathrm{kg}\) each. There were 89 more heavier boxes than lighter boxes, and the total weight of all the boxes was \(1,974.28 \mathrm{kg} .\) How many boxes were there altogether?

Perform the indicated operations and simplify as completely as possible. $$x(x-2 y)+y(x+y)$$

Perform the indicated operations on the given inequality. Sketch the resulting inequality on a number line. \(-x<3 ;\) multiply each side by \(-1\)

Perform the indicated operations on the given inequality. Sketch the resulting inequality on a number line. \(-x<3 ;\) divide each side by \(-1\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

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.