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Chapter 12: Problem 13

Graph each inequality. $$x \geq 4$$

Short Answer

Expert verified
Place a closed dot on 4 and shade to the right on the number line.

Step by step solution

01

Identify the Type of Inequality

This is an inequality that includes 'greater than or equal to'. Specifically, it is represented as:\[ x \text{ }\textgreater{or equal to}\text{ } 4\]
02

Draw the Number Line

Draw a horizontal number line and mark the point for 4 clearly. Label the points around 4 to give context, such as 3, 4, and 5.
03

Indicate the Boundary Point

Since the inequality includes 'equal to' (\(\text{鈮\)), place a closed dot on 4 on the number line to indicate that 4 is included in the solution set.
04

Shade the Solution Area

Shade the number line to the right of the number 4. This indicates that all numbers greater than or equal to 4 are included in the solution. The shading should extend indefinitely to the right.

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Key Concepts

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

Number Line
A number line is a visual tool used in mathematics to represent numbers in a linear format. It runs horizontally, with positive numbers found to the right of zero and negative numbers to the left.
The number line helps in identifying the relative positions of numbers and is essential for graphing inequalities.
For example, if you need to graph the inequality \(x \text{ } \geq \text{ } 4\), you will first draw a horizontal line and label points such as 3, 4, and 5 to give context.
Inequality
An inequality compares two values, showing if one is less than, greater than, or simply not equal to the other. Inequalities are represented using symbols:
  • \( < \): Less than
  • \( > \): Greater than
  • \( \leq \): Less than or equal to
  • \( \geq \): Greater than or equal to
In the inequality \(x \text{ } \geq \text{ } 4\), the symbol \( \geq \) indicates that \(x\) can be any number greater than or equal to 4.
Solution Set
The solution set of an inequality includes all the values that satisfy the inequality. For \(x \text{ } \geq \text{ } 4\), the solution set contains all numbers starting from 4 and extending to infinity in the positive direction.
When representing this on a number line, you'll use a closed dot on the number 4 to show that 4 is part of the solution set.
From the closed dot, you'll shade the line to the right to indicate all numbers greater than 4 are also part of the solution set.
Greater than or Equal to
The term 'greater than or equal to' is represented using the symbol \( \geq \). This symbol means that the value on the left is either greater than or exactly equal to the value on the right.
In our exercise, \(x \text{ } \geq \text{ } 4\) means \(x\) can be 4, or any number larger than 4.
On a number line, you represent this by placing a closed dot on the number 4 to show that 4 is included, and then shading to the right to include all numbers greater than 4.

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Most popular questions from this chapter

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Factor each of the following: (a) \(4(2 x-3)^{3} \cdot 2 \cdot\left(x^{3}+5\right)^{2}+2\left(x^{3}+5\right) \cdot 3 x^{2} \cdot(2 x-3)^{4}\) (b) \(\frac{1}{2}(3 x-5)^{-\frac{1}{2}} \cdot 3 \cdot(x+3)^{-\frac{1}{2}}-\frac{1}{2}(x+3)^{-\frac{3}{2}}(3 x-5)^{\frac{1}{2}}\)

A young couple has \(\$ 25,000\) to invest. As their financial consultant, you recommend that they invest some money in Treasury bills that yield \(7 \%,\) some money in corporate bonds that yield \(9 \%,\) and some money in junk bonds that yield \(11 \% .\) Prepare a table showing the various ways that this couple can achieve the following goals: (a) \(\$ 1500\) per year in income (b) \(\$ 2000\) per year in income (c) \(\$ 2500\) per year in income (d) What advice would you give this couple regarding the income that they require and the choices available?

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. If \(z=6 e^{i \frac{7 \pi}{4}}\) and \(w=2 e^{i \frac{5 \pi}{6}},\) find \(z w\) and \(\frac{z}{w} .\) Write the answers in polar form and in exponential form.

Find the function \(y=a x^{2}+b x+c\) whose graph contains the points \((1,-1),(3,-1),\) and (-2,14).

Solve each system of equations. If the system has no solution, state that it is inconsistent. $$ \left\\{\begin{array}{rr} x-2 y+3 z= & 7 \\ 2 x+y+z= & 4 \\ -3 x+2 y-2 z= & -10 \end{array}\right. $$
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