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Chapter 3: Problem 27

$$ \text { For Problems 23-60, solve each inequality. (Objective 1) } $$ $$ 4 x>36 $$

Short Answer

Expert verified
The solution is \( x > 9 \).

Step by step solution

01

Identify the Inequality

The inequality given is \( 4x > 36 \). Our goal is to find all values of \( x \) that satisfy this inequality.
02

Isolate the Variable

To isolate the variable \( x \), divide both sides of the inequality by 4. This operation is valid and maintains the direction of the inequality since we are dividing by a positive number.
03

Perform the Division

Divide \( 4x \) by 4 and divide 36 by 4: \[\frac{4x}{4} > \frac{36}{4} \]This simplifies to \( x > 9 \).
04

Write the Solution

The solution to the inequality is \( x > 9 \). This means that any number greater than 9 will satisfy the original inequality.

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Key Concepts

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

Isolating the Variable
In inequality problems, isolating the variable is crucial. This means making the variable the subject of the inequality, allowing us to clearly see its potential values. Think of it like turning a cluttered fact into a clearly written problem statement.
To achieve this, follow these steps:
  • Identify operations applied to the variable: Look at what's being done to the variable. In our example, the variable \( x \) is being multiplied by 4, noted in \( 4x > 36 \).
  • Reverse the operations: To isolate \( x \), we need to reverse the multiplication by doing its opposite, which is division.
    This helps in isolating \( x \) from the rest of the terms.
This systematic approach simplifies the process and the variable stands alone, reflecting the values it can take in solving the inequality.
Division in Inequalities
Division plays a key role in solving inequalities, especially when aiming to isolate the variable. It is essential to perform division accurately to maintain the inequality's integrity.
  • Divide using positive numbers: When dividing both sides of an inequality by a positive number, the direction of the inequality remains unchanged. In our example, dividing both sides by 4, which is positive, keeps the inequality as \( x > 9 \).
  • Beware of negative numbers: Dividing by a negative number flips the direction of the inequality.
    Always double-check the number you are dividing by to avoid mistakes.
Ensuring each step in division is correct prevents errors and helps clearly showcase the possible values for the variable.
Solution Set of Inequalities
The solution set represents all possible values that satisfy the inequality. Once the variable is isolated and division done correctly, understanding this set becomes simpler.
  • Define the solution range: For the inequality \( x > 9 \), the solution set includes all numbers greater than 9.
    This means 10, 11, and further beyond are included.
  • Use number lines: Visual representations can help. On a number line, draw an open circle at 9 and shade everything to the right to indicate that values greater than 9 are solutions.
This method provides a clear and visual way of comprehending the extent of solutions, reinforcing the understanding of the problem at hand.

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Most popular questions from this chapter

Solve each equation. $$3(x+1)+4(2 x-1)=5(2 x+3)$$

Solve each equation. $$5(x-3)=35$$

Mrs. Nelson had to wait 4 minutes in line at her bank's automated teller machine. This was 3 minutes less than one-half of the time she waited in line at the grocery store. How long in minutes did she wait in line at the grocery store?

Solve each equation. $$\frac{x-1}{4}+\frac{x+2}{5}=\frac{39}{20}$$

Solve each equation. $$\frac{x+2}{3}+\frac{x+3}{4}=\frac{13}{3}$$
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