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Chapter 3: Problem 29

$$ \text { For Problems 23-60, solve each inequality. (Objective 1) } $$ $$ 6 x<20 $$

Short Answer

Expert verified
The solution is \( x < \frac{10}{3} \).

Step by step solution

01

Understanding the Inequality

The inequality given is \( 6x < 20 \). Our goal is to solve for \( x \), which means we need to isolate \( x \) on one side of the inequality to find the values \( x \) can take.
02

Isolating x

To isolate \( x \), divide both sides of the inequality by 6. This is how we handle the coefficient attached to \( x \): \( \frac{6x}{6} < \frac{20}{6} \) Simplifying this gives us: \( x < \frac{10}{3} \)
03

Simplifying the Fraction

Further simplify the fraction \( \frac{20}{6} \) to its lowest terms by dividing the numerator and the denominator by their greatest common divisor, which is 2. This results in: \( x < \frac{10}{3} \)

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Key Concepts

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

Linear Inequalities
Linear inequalities are mathematical expressions that describe a range of values for variables similar to how linear equations do, but with inequality signs such as \(<, >, \leq, \geq\).
In the given exercise, you encountered the inequality \(6x < 20\).
This essentially means: find the range of values for \(x\) that satisfy this condition.To solve a linear inequality:
  • Apply similar principles as solving equations, but always remember the direction of the inequality sign matters.
  • Whenever you multiply or divide both sides by a negative number, the direction of the inequality reverses. This can be a point of confusion, but for this problem, such conditions do not apply as we're dividing by a positive number.
  • Keep the variable terms on one side and constants on the other until you isolate the variable.
Linear inequalities commonly appear in contexts involving constraints, like budgeting or maximizing resources, making them incredibly useful in real-world scenarios.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying expressions to either solve or better understand the forms they represent. In this exercise, manipulating the inequality \(6x < 20\) was key to isolate \(x\).
The primary tool was division, as we needed to undo the multiplication of \(x\) by 6:
  • Divide both sides of the inequality by the coefficient of \(x\), i.e., 6.
  • This operation simplifies to: \(\frac{6x}{6} < \frac{20}{6}\), which ultimately leads to \(x < \frac{10}{3}\).
This step-by-step division helps in keeping the inequality balance intact, provided you perform it on both sides.
It demonstrates the principle of balancing, indicating whatever operation you apply to one side must also be done on the other. This is foundational in pretty much all algebraic processes.
Fraction Simplification
Fraction simplification is the process of reducing the fraction to its simplest form by finding the greatest common divisor (GCD) of the numerator and the denominator.
In the solution to \(6x < 20\), we specifically simplified the fraction \(\frac{20}{6}\).Here are the steps to simplify fractions:
  • Identify the greatest common divisor of the numerator and the denominator. Here, it's 2 for \(20\) and \(6\).
  • Divide both the numerator and the denominator by the GCD, resulting in \(\frac{10}{3}\).
  • You now have the inequality \(x < \frac{10}{3}\).
This simple adjustment ensures the expression is as clear and concise as possible, making further calculations easier.
Fraction simplification is an essential skill, especially when dealing with results of division that are not immediately evident in their simplest form.

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Most popular questions from this chapter

Tracy received a bill with cell phone use charges of \(\$ 136.74\). Included in the \(\$ 136.74\) was a charge of \(\$ 39.99\) for the monthly plan and a charge for 215 extra minutes. How much is Tracy being charged for each extra minute?

Robin paid a \(\$ 454\) bill for a car DVD system. This included \(\$ 379\) for the DVD player and \(\$ 60\) an hour for installation. Find the number of hours it took to install the DVD system.

Solve each equation. $$-(x+7)=-2(x+10)$$

Solve each equation. $$5(x-4)=4(x+6)$$

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