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Chapter 1: Problem 40

Solve the inequality. Then graph the solution set on the real number line. \(3 x+1 \geq 2+x\)

Short Answer

Expert verified
The solution to the inequality \(3 x+1 \geq 2+x\) is \(x \geq 0.5\) and the graph representation would have a closed circle at 0.5, with shaded region extending to the right.

Step by step solution

01

Solving the Inequality

To solve \(3 x+1 \geq 2+x\), transfer \(x\) from the right-hand side to the left-hand side by subtracting \(x\) from both sides. You will get \(3x - x \geq 2 - 1\), which simplifies to \(2x \geq 1\). Next, to solve for \(x\), divide both sides by \(2\). Hence, \(x \geq 0.5\).
02

Visualizing on the Number Line

To represent \(x \geq 0.5\) on the number line, shade all the numbers greater than or equal to \(0.5\). This means that you will draw a closed circle at \(0.5\), and then shade to the right. The close circle at \(0.5\) indicates that \(0.5\) is included in the solution set because \(x\) is greater than or equal to \(0.5\).

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Key Concepts

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

Understanding the Real Number Line
The real number line is a fundamental concept in mathematics, serving as a visual representation of all possible numbers. It stretches infinitely in both directions, with each point on the line corresponding to a real number.
To better understand how the real number line works, imagine it as a straight line where:
  • Zero is positioned in the center.
  • Positive numbers are placed to the right of zero.
  • Negative numbers lie to the left of zero.
This visualization helps in comparing the sizes of numbers, determining greater or smaller values easily by their position on the line.
In the original exercise, we use the real number line to graph the solution of the inequality. This allows us to clearly see all the numbers that satisfy the inequality condition.
A Simple Guide to Graphing Inequalities
Graphing inequalities involves illustrating the set of possible solutions on the real number line. It's a practical way to view which numbers make the inequality true. Here is how you can graph an inequality, using the original exercise as a reference:
  • First, solve the inequality algebraically, which we have done to get the solution, \(x \geq 0.5\).
  • On the real number line, find the point corresponding to \(0.5\). Since the inequality includes \(0.5\), denote this with a closed circle.
  • Shade all points to the right side of \(0.5\). This shading indicates that all numbers greater than \(0.5\) are part of the solution set.
This visual method is not only efficient, but can also be applied to a variety of inequalities, helping you immediately identify all the possible solutions. With practice, interpreting these graphs will become second nature.
Exploring Algebraic Expressions
Algebraic expressions are mathematical phrases that can contain numbers, variables, and operation symbols. They are the building blocks of algebra and are used to represent relationships and conditions, such as inequalities.
In the original exercise, we used the algebraic expression \(3x + 1 \geq 2 + x\). Let's break this down:
  • The terms \(3x\) and \(x\) represent parts of the expression that include the variable \(x\).
  • The constant numbers \(1\) and \(2\) are standalone numbers added to the terms.
  • The inequality symbol \(\geq\) signifies that one side is "greater than or equal to" the other.
To solve algebraic expressions like this, we apply basic algebraic operations such as addition, subtraction, and division. The goal is to isolate the variable on one side of the inequality to find its possible values. This approach simplifies expressions and makes it easier to analyze and solve mathematical problems.

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Most popular questions from this chapter

Solve the inequality. Then graph the solution set on the real number line. \(x^{2}+x<6\)

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