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Chapter 3: Problem 17

Solve each inequality. $$5 x+3 \leq 8$$

Short Answer

Expert verified
x 鈮� 1

Step by step solution

01

Subtract 3 from both sides

To isolate the term with the variable, subtract 3 from both sides of the inequality. This gives: \[ 5x + 3 - 3 \leq 8 - 3 \] which simplifies to: \[ 5x \leq 5 \]
02

Divide both sides by 5

Next, divide both sides of the inequality by 5 to solve for x. This gives: \[ \frac{5x}{5} \leq \frac{5}{5} \] which simplifies to: \[ x \leq 1 \]

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Key Concepts

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

Elementary Algebra
Elementary algebra is a branch of mathematics that deals with variables and the rules for manipulating these variables. It is the foundational part of algebra and is used extensively in various fields.

Let's break it down:

  • Variables: Symbols, often letters, that represent unknown values.
  • Constants: Fixed values, like numbers.
  • Operations: Actions like addition, subtraction, multiplication, and division.
In elementary algebra, you'll learn to work with equations and inequalities. Understanding these basics helps in solving more complex problems later on. The primary goal is to isolate the variable to find its value, as shown in the step-by-step solution you provided.
Isolating Variable
Isolating the variable is one of the first and most critical skills you'll need in algebra. It involves manipulating the equation or inequality so that the variable stands alone on one side.

Let's understand the process using the given example:

Inequality: \(5x + 3 \leq 8\)
  • Step 1: Subtract 3 from both sides of the inequality. The goal is to eliminate the constant term (3) on the left side:

    • \(5x + 3 - 3 \leq 8 - 3 \)


    \(5x \leq 5 \)


  • Step 2: Divide both sides by 5 to isolate x. This reduces the term to its simplest form:

    • \(\frac{5x}{5} \leq \frac{5}{5} \)


    \(x \leq 1 \)


The variable x is now isolated. Always remember to perform the same operation on both sides to keep the equation balanced.
Linear Inequalities
Linear inequalities are similar to linear equations but with inequality signs instead of an equal sign. Inequalities show that one side is not necessarily equal to the other but could be greater, lesser, or equal depending on the sign used.

Here are the types of inequality signs you'll encounter:
  • \(\leq\) : Less than or equal to
  • \(\geq\) : Greater than or equal to
  • \(<\) : Less than
  • \(>\) : Greater than
When solving linear inequalities:
  • Treat it like an equation when performing arithmetic operations.
  • Always reverse the inequality sign if you multiply or divide by a negative number.
For the given exercise, \(5x + 3 \leq 8\), you followed standard steps to isolate x, resulting in \(x \leq 1\).
Step-by-Step Solution
Breaking down each step helps in better understanding and avoiding mistakes. Here is your step-by-step solution:
  • Step 1: Subtract 3 from both sides. This eliminates the constant on the left side:

    • \(5x + 3 - 3 \leq 8 - 3 \) simplifies to \(5x \leq 5 \)


  • Step 2: Divide both sides by 5 to isolate x:

    • \(\frac{5x}{5} \leq \frac{5}{5} \) simplifies to \(x \leq 1 \)


Always double-check each step to ensure accuracy. Following steps methodically guarantees that you solve the inequality correctly.

Understanding each of these steps and the rationale behind them is key to mastering algebra. Practice different problems to get comfortable with the process.

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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. Find the length of a rectangle whose width is 3.5 yd and whose area is 73.5 sq yd.

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?

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 trucking company determines that the cost \(C\) (in dollars per mile) of operating a truck is given by \(C=0.003 s+0.21,\) where \(s\) is the average speed of the truck. (a) Find the cost per mile if the truck averages 55 miles per hour. (b) Find the average speed that yields a cost per mile of \(\$ 0.35\)

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

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?
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