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Chapter 5: Problem 9

Graph each inequality. $$ x \leq 1 $$

Short Answer

Expert verified
The graph of the inequality \(x \leq 1\) is a number line with a closed circle at point 1 and all the values to the left of this point shaded or highlighted.

Step by step solution

01

Understand the inequality

The inequality \(x \leq 1\) means any value of x is less than or equal to 1. Therefore, the number line should represent all these values.
02

Draw the number line

To represent this, begin with drawing a simple number line. Place the value of 1 on the number line as the reference point.
03

Identify and indicate the point on the number line

Indicate the point '1' on the number line with a closed circle to represent that x can be equal to 1.
04

Shade the values of x

Since x can be anything less than or equal to 1, all values to the left of 1 on the number line should be shaded or highlighted. This shows that any value in the shaded region, including 1, is a valid solution for the given inequality.

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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 representation that helps us understand numerical relationships and inequalities. Imagine it as a straight line where each point corresponds to a number.
The line typically includes both negative and positive numbers, centered around zero. To graph inequalities, the number line is an invaluable tool. It provides a clear picture of values relative to each other.
In our exercise, we needed to draw a number line to represent the inequality \( x \leq 1 \). This means we'll focus on values surrounding and including the number 1. By marking numbers from a chosen range, we create a handy reference to visualize the inequality.
Inequalities
Inequalities are mathematical statements that compare two values using inequality symbols such as \( <, >, \leq, \geq \). These symbols tell us how one value is related to another.
For instance, the inequality \( x \leq 1 \) expresses that \( x \) can be 1, or any number less than 1. Unlike equalities, inequalities offer a range of solutions rather than one definite answer.
When working with inequalities, always pay attention to the symbol used. If there's a line under the arrow, like in \( \leq \) or \( \geq \), it indicates that the endpoint itself is included in the solution. Understanding these symbols helps in accurately graphing solutions on a number line.
Shading Regions
When graphing inequalities on a number line, shading is used to show the valid solutions. This technique highlights the section of the number line that satisfies the inequality.
For example, in the inequality \( x \leq 1 \), we shade all the numbers that are less than or equal to 1, which means we're shading the entire left side of the number line from 1.
Closed circles are used to denote that the endpoint, in this case the number 1, is included in the solution as specified by the \( \leq \) symbol. Think of the shaded region as saying, "Here are the values of \( x \) that work for the inequality!"
Through shading, not only do we show the solutions, but also communicate visually how inclusive inequalities operate.

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In Exercises \(5-18\), solve each system by the substitution method. $$ \begin{aligned} &4 x+3 y=0\\\ &2 x-y=0 \end{aligned} $$

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