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Chapter 4: Problem 14

In Exercises \(13-16\), sketch the graph of the linear inequality. $$ x>-4 $$

Short Answer

Expert verified
The graphical representation of the inequality \(x > -4\) is an open circle at -4 on the number line, with everything to the right of this point shaded in to represent all values of x that are greater than -4.

Step by step solution

01

Understand the Inequality

The inequality presented is \(x > -4\). This inequality says that x, which is the value we are interested in, is greater than -4.
02

Draw a Number Line

Draw a straight line, which is going to be called the number line. The number line is a graphical representation of all real numbers. Each number corresponds to a point on the number line. Mark the number -4 on the number line as this is the value given in the inequality.
03

Represent the Solution Set

On the line, any point to the right of -4 represents x-values greater than -4. This represents the solution set of the inequality \(x > -4\). Because the inequality doesn't include -4 (it鈥檚 just greater than -4, not greater than or equal to), we use an open circle at -4 to show that -4 isn鈥檛 part of the solution.
04

Shade the Solution Set

Shading is a method to represent all the number that satisfy an inequality on the number line. Shade everything to the right of the open circle at -4. This shading represents all values of x that are greater than -4.

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Key Concepts

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

Inequality Representation
Understanding how to represent inequalities is crucial for solving algebraic problems effectively. Inequalities such as \(x > -4\) describe a range of values rather than a specific number. This particular inequality tells us that the variable \(x\) can be any value greater than -4, but not -4 itself.

The 'greater than' symbol (>) in \(x > -4\) indicates that we are dealing with all numbers to the right of -4 on a number line. Note that had the inequality been \(x \geq -4\) (greater than or equal to), it would also include -4 as part of the solution set. Understanding these symbols and their implications is fundamental to correctly graphing the solutions.
Number Line
The number line is a foundational tool in mathematics that visually represents the ordering of numbers. It's a horizontal line with marks at regular intervals, each representing a different number. Think of it like a ruler for numbers. To graph an inequality like \(x > -4\), you'll first draw a straight horizontal line and mark the point -4.

Remember, every point on the line to the left represents numbers less than -4, while points to the right represent numbers greater than -4. By using a number line, you can easily show the range of possible values for \(x\) that satisfy the inequality.
Solution Set Shading
After plotting an inequality on a number line, the next step is to indicate the solution set. Shading is a common technique used to highlight all the numbers that solve the inequality. For the inequality \(x > -4\), you would begin by drawing an open circle at -4 because -4 is not part of the solution. It's open to show exclusion.

Next, shade the entire portion of the number line to the right of -4. This shaded area visually represents all the values of \(x\) that meet the criteria of being greater than -4. It's a simple yet powerful way to communicate the range of numbers that form the solution to an inequality.

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Most popular questions from this chapter

In Exercises 71-76, write an equation of the line that passes through the points. $$ (0,0),(-3,6) $$

In Exercises 47-52, write the statement as a linear inequality. Then sketch the graph of the inequality. The sum of \(x\) and \(y\) is at least 9 .

In Exercises \(13-16\), sketch the graph of the linear inequality. $$ x \leq 3 $$

In Exercises 65-68, determine the slope of the line passing through the points. $$ (-1,3) \text { and }(2,-5) $$

In Exercises 71-76, write an equation of the line that passes through the points. $$ (5,0),(0,-2) $$
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