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Chapter 2: Problem 301

Graph each inequality on the number line and write in interval notation. (a) \(-2

Short Answer

Expert verified
(a) (-2, 0); (b) [-5, -3); (c) [0, 3.5].

Step by step solution

01

- Graph Inequality (a)

To graph of the number line: 1. Draw an open circle at -2, because -2 is not included in the inequality. 2. Draw another open circle at 0, because 0 is also not included in the inequality. 3. Shade the region between -2 and 0 to represent all numbers between -2 and 0. 4. The interval notation for this inequality is (-2, 0).
02

- Graph Inequality (b)

To graph of the number line: 1. Draw a closed circle at -5, because -5 is included in the inequality. 2. Draw an open circle at -3, because -3 is not included in the inequality. 3. Shade the region between -5 and -3 to represent all numbers between -5 and -3. 4. The interval notation for this inequality is [-5, -3).
03

- Graph Inequality (c)

To graph of the number line: 1. Draw a closed circle at 0, because 0 is included in the inequality. 2. Draw another closed circle at 3.5, because 3.5 is also included in the inequality. 3. Shade the region between 0 and 3.5 to represent all numbers between 0 and 3.5. 4. The interval notation for this inequality is [0, 3.5].

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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 straight line that represents numbers in increasing value from left to right. It is a useful tool for visualizing and solving inequalities. Each point on the line corresponds to a real number, and it helps us understand the relationship between different numbers and intervals.

For example, to graph the inequality \(-2 \lt x \lt 0\), you would:
  • Locate -2 and 0 on the number line.
  • Use open circles at -2 and 0 because these exact values are not included (only the numbers between them are).
  • Shade the region between -2 and 0 to represent that x can be any number between -2 and 0.
This makes it clear which parts of the number line satisfy the inequality.
interval notation
Interval notation is a way of writing subsets of the real number line. It's compact and conveys both the start and end points of an interval along with whether the endpoints are included.

  • Open intervals: Both endpoints are not included and are represented with parentheses, e.g., (a, b).
  • Closed intervals: Both endpoints are included, represented with brackets, e.g., [a, b].
  • Half-open intervals: One endpoint is included, and the other is not, e.g., [a, b) or (a, b].
For the inequality \(-2 \lt x \lt 0\), the interval notation is (-2, 0). This tells us that x includes all numbers between -2 and 0, but not -2 and 0 themselves.
closed circle
In graphing inequalities on a number line, a closed circle indicates that a particular number is included in the solution set.

For example, in the inequality \(-5 \leq x \lt -3\):
  • Draw a closed circle at -5 because -5 is included in the solution (indicated by \(\leq\)).
  • Draw an open circle at -3 because -3 is not included (indicated by \(\lt\)).
  • Shade the region between -5 and -3 to show all the numbers that x can be.
The closed circle at -5 tells us that -5 is a boundary point and is included within the interval.
open circle
In graphing inequalities, an open circle means that a particular value is not included in the solution set. Open circles help differentiate numbers that are boundaries but not part of the actual solution.

For example, consider the inequality \(-2 \lt x \lt 0\):
  • Draw an open circle at -2 and another open circle at 0.
  • Shade the area between -2 and 0 to indicate all numbers between -2 and 0 are part of the solution.
The open circles at -2 and 0 make it clear that x does not actually equal -2 or 0, just any number in between.

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Most popular questions from this chapter

Solve. $$ |2 x-3|-4=1 $$

Solve each inequality, graph the solution on the number line, and write the solution in interval notation. $$ \frac{3}{4} x-2>4 \text { or } 4(2-x)>0 $$

Solve each inequality, graph the solution, and write the solution in interval notation. $$ x \leq 4 \text { and } x>-2 $$

Solve each inequality, graph the solution on the number line, and write the solution in interval notation. $$ x \leq-4 \text { or } x>-3 $$

Solve. $$ |3 x-2|=|2 x-3| $$
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