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Chapter 17: Problem 42

Graph the given inequalities on the number line. $$x<1 \text { or } 1

Short Answer

Expert verified
Graph two regions: (x < 1) with an open circle at 1, and (1 < x 鈮� 4) with an open circle at 1 and a closed circle at 4.

Step by step solution

01

Understand the Inequalities

The problem gives two inequalities: \(x < 1\) and \(1 < x \leq 4\). These inequalities describe the ranges of \(x\) that we need to mark on a number line.
02

Identify the Critical Points

The critical points in these inequalities are where the inequality expression changes. In our case, these points are \(1\) and \(4\).

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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 visual tool to represent numbers in a straight line where each point corresponds to a real number. It's commonly used to represent inequalities graphically. Imagine the number line as a ruler, where each tick mark represents an integer. Numbers to the right are greater than those to the left.

When dealing with inequalities, we use open or closed circles to signify which numbers are included in the solution set:
  • An open circle means the number is not part of the solution. This is used for inequalities like <, or >.
  • A closed circle means the number is included in the solution, used for 鈮� or 鈮�.
By plotting the corresponding numbers for each inequality, you can easily visualize the range of solutions.
Critical Points
Critical points play a vital role in graphing inequalities as they signal where the inequality expression changes. They serve as boundaries or endpoints for the solution set.

In the given exercise, the critical points are 1 and 4:
  • The number 1 separates the inequality into two distinct parts: where values are less than 1, and where values are greater than 1.
  • The number 4 is crucial because it marks the endpoint of the second inequality, with values up to and including 4.
By locating these points on the number line, you can clearly define where the inequalities start and stop, helping you to accurately graph the solution.
Inequalities
Inequalities are mathematical expressions that denote the relationship between two values. In this exercise, you work with two types of inequalities:

1. **`x < 1`** - This inequality specifies that any number less than 1 is part of the solution.
2. **`1 < x 鈮� 4`** - This inequality indicates numbers greater than 1, but also includes numbers up to and including 4.

When graphing these inequalities, it's essential to differentiate between strict inequalities (< and >) and non-strict inequalities (鈮� and 鈮�), as they determine whether endpoints are part of the solution. This distinction is important for correct graph representation.
Mathematical Ranges
Mathematical ranges reflect intervals on the number line that contain all solutions of an inequality. These ranges can be finite or infinite, depending on the nature of the inequalities.

In the context of this problem, the ranges are:
  • Range for `x < 1`: This range includes all numbers to the left of 1, going infinitely leftward.
  • Range for `1 < x 鈮� 4`: This range begins just after 1 and includes all numbers up to and including 4.
Understanding these ranges helps to clearly visualize spans that answer the inequality on the number line. Proper comprehension of these ranges ensures accurate plotting and interpretation of the solutions on a graph.

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Most popular questions from this chapter

Solve the given problems. Solve for \(x\) if \(|x-1| > 4\) and \(|x-3| < 5\).

Solve the given problems. Solve for \(x: 1<|x-2| < 3\).

Answer the given questions by solving the appropriate inequalities. A triangular postage stamp is being designed such that the height \(h\) is \(1.0 \mathrm{cm}\) more than the base \(b\). Find the possible height \(h\) such that the area of the stamp is at least \(3.0 \mathrm{cm}^{2}\)

Determine the values for \(x\) for which the radicals represent real numbers. $$\sqrt{x^{2}-3 x}$$

Use a calculator to display the solution of the given inequality or system of inequalities. $$2 x+y<5$$
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