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Chapter 2: Problem 33

Graph each inequality. $$ x+3 \geq 0 $$

Short Answer

Expert verified
Shade right of -3 including -3 on the number line.

Step by step solution

01

- Solve the Inequality

First, solve the given inequality for the variable x. Start by isolating x:\[ x + 3 \geq 0 \]Subtract 3 from both sides:\[ x \geq -3 \]
02

- Draw the Number Line

Draw a number line to represent the solution. Locate -3 on the number line.
03

- Shade the Solution Area

Since the inequality is \( x \geq -3 \), shade the number line to the right of -3, including -3. Use a solid dot at -3 to indicate that -3 is included in the solution.
04

- Verify the Solution

Check any point in the shaded region to ensure it satisfies the inequality. For instance, if \( x = 0 \): \[ 0 + 3 = 3 \geq 0 \] This is true, confirming the solution is correct.

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Key Concepts

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

Inequalities
Inequalities are mathematical expressions that show the relationship between two values. Unlike equations where two sides are equal, inequalities use symbols like >, <, 鈮�, and 鈮�. In this particular exercise, we use the 'greater than or equal to' symbol (鈮�). The goal is to find all the values of x that make the inequality true. For example, if we have the inequality: \( x + 3 \geq 0 \),we want to isolate x. Subtracting 3 from both sides, we get: \( x \geq -3 \). So, any value of x that is -3 or greater will satisfy this inequality.
Number Line
A number line is a visual representation of numbers in a straight line. It helps in understanding the range of solutions for an inequality. To draw a number line for the inequality \( x \geq -3 \),
  • First, draw a horizontal line and mark points at regular intervals.
  • Locate the point -3 on the line.
  • Use a solid dot to indicate that -3 is included in the solution.
This gives a visual clue that -3 is part of the solution set.
Shading Solution Area
Shading the solution area helps to illustrate where the values that satisfy the inequality lie. For \( x \geq -3 \):
  • Start at -3.
  • Shade all the numbers to the right of -3 on the number line. This indicates that any number greater than or equal to -3 is part of the solution.
This shading is crucial as it visually distinguishes the range of all possible solutions. Always remember to check a point in the shaded area to ensure it satisfies the original inequality. For instance, if we test x = 0 in: \( 0 + 3 = 3 \geq 0 \)we find the solution holds true.

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Most popular questions from this chapter

An equation that defines \(y\) as a function \(f\) of \(x\) is given. (a) Solve for \(y\) in terms of \(x\), and write each equation using function notation \(f(x) .\) (b) Find \(f(3)\). $$ y-3 x^{2}=2 $$

An equation that defines \(y\) as a function \(f\) of \(x\) is given. (a) Solve for \(y\) in terms of \(x\), and write each equation using function notation \(f(x) .\) (b) Find \(f(3)\). $$ -2 x+5 y=9 $$

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