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Chapter 7: Problem 17

Graph. $$x<4$$

Short Answer

Expert verified
The graph of the inequality \(x<4\) is a number line with an open circle at 4 and a line extending to the left of 4.

Step by step solution

01

Identify inequality characteristics

Firstly, identify that this is a strict inequality. This means that the values of \(x\) are strictly less than 4 but does not include 4.
02

Create a number line

Then create a number line, placing zero at the center with regular interval markings on the number line. Mark the point 4 on the number line.
03

Represent the solution on the number line

Because \(x\) is less than 4 but not equal to, place an open circle at 4 on the number line. Then shade or draw a line to the left of 4, extending to the end of the number line. This shows that all values less than 4 are in the solution set.

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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 simple yet powerful tool used in mathematics to visualize numbers and their relationships. It consists of a straight line on which every point corresponds to a real number. Typically, the number line extends infinitely in both directions, but when graphing inequalities like \(x<4\), you only need to focus on a section of it.
  • The number 0 is usually placed at the center, with positive numbers to the right and negative numbers to the left.
  • When graphing an inequality, it's crucial to mark the significant values. In this case, we mark the number 4.
A number line helps students see where each value belongs and how they compare to each other, particularly useful in illustrating inequalities.
Strict Inequality
In mathematics, a strict inequality occurs when one value is strictly less than or greater than another. The symbol "less than" (\(<\)) indicates that the value on the left is not equal to, but smaller than the value on the right.
  • For the inequality \(x<4\), any value of \(x\) must be less than 4, and 4 itself is not part of the solution.
  • To communicate that 4 is not included, we use an open circle on the number line at this point.
Strict inequalities represent a range of numbers but exclude the boundary numbers. This exclusion is key to understanding the concept and differentiates it from non-strict (or "less than or equal to") inequalities, which include the boundary number.
Solution Set
The solution set of an inequality is a collection of all possible values that satisfy the inequality. For \(x<4\), the solution set consists of all real numbers less than 4. This is visually depicted on the number line.
  • You start at 4 with an open circle, signifying that 4 is not included in the set.
  • Then, you draw a shaded line extending to the left, covering all numbers less than 4.
This graphical representation provides a clear visual of the infinite set of numbers that make up the solution set of the inequality. It helps learners see that there is not just one solution, but rather a multitude of values that are part of the satisfying set.

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Most popular questions from this chapter

Graph each inequality for nonnegative values of \(x\) and \(y\). Then give some ordered pairs that satisfy the inequality. SEE EXAMPLE \(5 .\) (OBJECTIVE 3) It costs a bakery \(3\)dollar to make a cake and \(4\)dollar to make a pie. Production costs cannot exceed \(120\)dollar per day. Find an inequality that shows the possible combinations of cakes, \(x,\) and pies, \(y,\) that can be made, and graph it in the illustration.

Graph each inequality. SEE EXAMPLE 2. (OBJECTIVE 2) $$y \leq x+2$$

Use elimination to solve each system. $$\left\\{\begin{array}{l}4(x+2 y)=15 \\\x+2 y=4\end{array}\right.$$

Explain how to find the boundary for the graph of an inequality.

Graph the solution set of each system of inequalities, when possible. SEE EXAMPLE 7. (OBJECTIVE 4) $$\left\\{\begin{array}{l} x \geq-1 \\ y>-2 \end{array}\right.$$
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