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Chapter 2: Problem 19

Graph each inequality and describe the graph using interval notation. $$ x<5 $$

Short Answer

Expert verified
The graph is a line with an open circle at 5 and shaded to the left, represented as \((-\infty, 5)\) in interval notation.

Step by step solution

01

Understand the Inequality

The inequality given is \( x < 5 \). This means that \( x \) can be any value that is less than 5. It does not include 5 itself.
02

Determine the Boundary and Interval

The boundary for this inequality is \( x = 5 \), but since \( x < 5 \), the number 5 is not included in the solution set. In interval notation, we use a parenthesis to show that 5 is not included: \( (-\infty, 5) \).
03

Graph the Inequality on a Number Line

To graph \( x < 5 \) on a number line, draw a line with numbers and highlight the portion of the line that represents numbers less than 5. Place an open circle at 5 to indicate that this value is not included in the solutions.

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Key Concepts

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

Number Line
When working with inequalities like \( x < 5 \), the number line is a crucial visual tool. Imagine a simple straight line with evenly spaced marks representing numbers. This representation helps us see which numbers satisfy the inequality. For \( x < 5 \), we focus on numbers smaller than 5.To graph \( x < 5 \):
  • Draw a straight horizontal line.
  • Mark key numbers along this line, including 0 and 5 since 5 is our boundary.
  • Highlight the part of the line that includes numbers less than 5. You'll shade or darken the line extending to the left side of the number 5.
This visual representation quickly shows the possible values "x" can take.
Interval Notation
Interval notation provides a concise way to describe a set of numbers, like the solutions to an inequality. For \( x < 5 \), the interval notation tells us the range of values that "x" can be.Here's how it works:
  • "-鈭�" represents all numbers smaller than any visible number, used for values that decrease indefinitely.
  • The number 5, our boundary, is written on the "right" of the interval since that's the upper limit for potential values of "x".
  • Because "x" cannot be exactly 5, we use a parenthesis: \((-\infty, 5)\). A parenthesis means "not including" the boundary number. It shows precisely where the number set stops, without including the endpoint.
Interval notation makes it simple to understand the solution set without confusion.
Open Circle
In inequalities like \( x < 5 \), an open circle plays an important role in graphing on the number line. It helps signal that the boundary value is not part of the solution set.Here's how to use it:
  • When graphing \( x < 5 \), after drawing your number line, find the location of the number 5.
  • Place a circle right above 5 on the number line. This circle indicates the end of the line you will shade.
  • Keep it open (not filled in) to show that while we go up to this point, we do not include 5 as part of the solutions.
Using an open circle ensures anyone reading the graph knows exactly where the solution set starts and stops without confusion.

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Most popular questions from this chapter

Solve each inequality or compound inequality. Write the solution set in interval notation and graph it. $$ 3(3-x) \geq 6+x $$

Solve each inequality. Write the solution set in interval notation and graph it. $$ 2(5 x-6)>4 x-15+6 x $$

Solve each equation and inequality. Write the solution set of each inequality in interval notation and graph it. a. \(7(a-3)<2(5 a-8)\) b. \(7(a-3)=2(5 a-8)\)

Solve each inequality or compound inequality. Write the solution set in interval notation and graph it. $$ 4 \leq-4(x-2)<20 $$

Solve \(B=R-\frac{1}{16}(c-3 D)\) for \(D\).
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