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Chapter 2: Problem 29

Solve each inequality. Write the solution set in interval notation and graph it. $$ \frac{2}{3} x \geq 2 $$

Short Answer

Expert verified
The solution set is \([3, \infty)\) and it is graphed with a closed dot at 3, shaded to the right.

Step by step solution

01

Understand the Inequality

The inequality given is \( \frac{2}{3} x \geq 2 \). This inequality wants us to find all values of \( x \) such that the expression on the left is greater than or equal to the number 2.
02

Isolate the Variable

To solve for \( x \), we first need to isolate it by getting rid of the fraction. Multiply both sides of the inequality by the reciprocal of \( \frac{2}{3} \), which is \( \frac{3}{2} \). This operation will give us \( x \geq 2 \times \frac{3}{2} \).
03

Simplify the Expression

Now, simplify the right side of the inequality: \( x \geq 3 \). This indicates that \( x \) must be greater than or equal to 3.
04

Write in Interval Notation

The solution \( x \geq 3 \) in interval notation is \( [3, \infty) \). This notation means that \( x \) can be any number including 3 and going to infinity.
05

Graph the Solution

On a number line, graph \( x \geq 3 \) by placing a closed dot or a bracket at 3 to indicate that 3 is included in the solution set, and shading the line to the right towards infinity.

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Key Concepts

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

Interval Notation
Interval notation is a way of representing solution sets for inequalities in a compact form. It uses brackets and parentheses to describe a range of values that satisfy the given condition.
The brackets used in interval notation hold specific meanings:
  • Square brackets [ ] indicate that an endpoint is included in the interval. For example, [3, 鈭�) means the value 3 is part of the solution set.
  • Parentheses ( ) indicate that an endpoint is not included. For instance, (3, 鈭�) would suggest that values start just past 3 and continue onward.
In the context of our exercise, the inequality \(x \geq 3\) is represented as \([3, \infty)\). This notation tells us that the solution includes the number 3 and all numbers greater than 3. The use of infinity (鈭�) signifies that there is no upper limit to the values x can take.
Graphing Solutions
Graphing solutions to inequalities visually demonstrates which values satisfy the inequality. A number line is often used to depict these solutions, making it clear and easy to understand.
  • To graph \(x \geq 3\), start by locating the number 3 on the number line. Place a closed dot or bracket at 3, since \(x\) is allowed to be equal to 3. This signifies inclusion of the number.
  • Next, shade the line extending to the right from 3. The shaded region shows that all numbers greater than or equal to 3 are part of the solution.
Graphing provides an immediate visual representation of an inequality solution that complements the numerical solution in interval notation. This method helps in quick evaluations of valid x values at a glance.
Solving Algebraic Expressions
Solving algebraic expressions within inequalities involves several steps to isolate the variable. Here, we tackle the inequality \(\frac{2}{3} x \geq 2\). The goal is to get \(x\) on one side by itself.
  • Identify the inequality: We start with \(\frac{2}{3} x \geq 2\), which asks what x values make the expression greater than or equal to 2.
  • Eliminate fractions: Multiply the entire inequality by the reciprocal of the fraction \(\frac{2}{3}\), which is \(\frac{3}{2}\). This makes the left side become just \(x\), simplifying our work. Be mindful to perform the same operation across both sides of the inequality.
  • Simplify: Carry out the multiplication on the right side: \(x \geq 2 \times \frac{3}{2}\), simplifying to \(x \geq 3\). This tells us that x must be greater than or equal to 3.
Each step carefully brings attention to maintaining the balance requirement intrinsic to solving equations and inequalities, which is ensuring that operations applied to one side are mirrored on the other. This process is crucial to arrive at an accurate solution for the inequality.

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