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Chapter 2: Problem 1

For exercises 1-8, use a number line graph to represent the inequality. $$ x<8 $$

Short Answer

Expert verified
Shade the number line to the left of an open circle at 8.

Step by step solution

01

Understand the Inequality

The inequality given is \(x < 8\). This means that \(x\) can be any value less than 8.
02

Draw a Number Line

Draw a horizontal line and mark numbers on it in equal intervals. Make sure to include the number 8 and several numbers on either side of it for context.
03

Mark the Key Point

Locate the number 8 on the number line. Since \(x < 8\), this number (8) will not be included in the solution set. Represent this by drawing an open circle around 8.
04

Shade the Solution Region

Shade the number line to the left of the open circle at 8. This indicates that all numbers less than 8 are solutions to the inequality.

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Key Concepts

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

number line representation
When we talk about number line representation, we're essentially using a visual tool to make abstract math concepts clearer. A number line is a straight, horizontal line on which we can mark numbers at equal intervals. This helps us easily see the relationship between different numbers.

Drawing a number line for inequalities involves a few key steps:
  • First, draw a straight horizontal line.
  • Next, mark equal intervals along the line and label them with numbers.
  • You can choose any range, but make sure to include the key numbers mentioned in the problem.
In this exercise, we are particularly interested in representing the inequality where 8 is a critical number. We'll focus on the value 8 and the numbers around it so we can show the solution clearly.
graphing inequalities
Graphing inequalities on a number line helps us visualize which numbers satisfy the inequality. Let's break down how to do this for the given inequality in the exercise, which is \(x < 8\).

Firstly, recognize what the inequality means. Here, \(x < 8\) tells us that \(x\) can be any number less than 8.
  • Draw your number line and mark the key number, which is 8.
  • Since \(x\) has to be less than 8, we need to show that every number to the left of 8 is a solution.
  • You shade the line to the left of 8.
Make sure to extend the shading well into the negative numbers so it鈥檚 clear that any smaller number meets the inequality.
open circles in inequalities
Understanding open circles is important for graphing inequalities correctly. The open circle indicates that a number is not included in the solution set. Let's examine this with the example given: \(x < 8\).
  • Find the number 8 on the number line.
  • Since \(x\) must be less than, but not equal to, 8, we put an open circle on 8.
The open circle means that 8 itself isn鈥檛 a solution, but numbers just a tiny bit smaller than 8 are. This visually shows that 8 does not satisfy the inequality, although numbers very close to 8 do.
Remember, if the inequality were \(\leq\) 8, we would use a closed circle to include the number itself.

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