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Chapter 3: Problem 58

Explain in words what the inequality \(2

Short Answer

Expert verified
The inequality \(2 < x \leq 5\) means \(x\) is between 2 and 5, excluding 2 but including 5.

Step by step solution

01

Identify the Inequality Components

The inequality \(2 < x \leq 5\) has two parts. The first part, \(2 < x\), means that the value of \(x\) must be greater than 2. The second part, \(x \leq 5\), means that \(x\) must be less than or equal to 5.
02

Combine the Conditions

When you combine \(2 < x\) and \(x \leq 5\), it means that \(x\) must be in the range between 2 and 5. However, \(x\) cannot be exactly 2, but it can be exactly 5.
03

Interpret the Inequality

The inequality \(2 < x \leq 5\) means that \(x\) is any real number greater than 2 but at most 5. In other words, \(x\) can take any value starting just above 2 and up to and including 5.

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Key Concepts

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

greater than
The phrase 'greater than' is a fundamental concept in algebra. When we see an inequality like \(2 Here are some examples:
  • If \(x = 3\), then \(2<3\) which is true.
  • If \(x = 2.1\), then \(2<2.1\) which is true.
  • If \(x = 2\), then \(2<2\) which is false because \(x\) is not greater than 2.

This kind of inequality defines a range of numbers that are all bigger than the given number. In our case, \(x\) could be 2.001, 3, or even 100. But it can never be 2 or less.
less than or equal to
The concept of 'less than or equal to' is represented by the symbol \(\leq\). When you see an expression like \(x \leq 5\), it means that the value of \(x\) can be any number up to and including 5.
Let's look at some examples:
  • If \(x = 5\), then \(x \leq 5\) which is true.
  • If \(x = 4\), then \(4 \leq 5\) which is also true.
  • If \(x = 6\), then \(6 \leq 5\) which is false because 6 is not less than or equal to 5.

In contrast to 'greater than', 'less than or equal to' allows the value to be exactly equal to the number specified. This broadens the range of acceptable values for \(x\).
range of values
A range of values describes a set of numbers that fit within certain bounds. In the inequality \(2 < x \leq 5\), the range of values for \(x\) is all the numbers that are greater than 2 and less than or equal to 5.
Let's break this down:
  • Starting just above 2, \(x\) can be 2.0001, 3, 4, and 5.
  • However, \(x\) cannot be 2, because \(2 < x\), and it cannot be more than 5, since \(x \leq 5\).

This means \(x\) is a real number within the interval (2, 5]. By understanding the range of values, you can solve inequalities and apply the right conditions to your solutions. Ranges are very useful in defining where solutions exist and where they do not.

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Most popular questions from this chapter

Solve each of the problems algebraically. That is, set up an equation and solve it. Be sure to clearly label what the variable represents. Round your answer to the nearest tenth where necessary. As dry air rises, it expands owing to the lower atmospheric pressure, and as a result the air cools at the rate of approximately \(5.4^{\circ} \mathrm{F}\) for each \(1,000\) feet of increase in altitude (up to an altitude of approximately \(40,000 \mathrm{ft}\) ). Thus, if the ground-level temperature is \(46^{\circ} \mathrm{F}\), the temperature at an altitude of \(A\) feet above the ground is given by the equation $$ T=46-0.0054 A $$ (a) Determine the temperature at an altitude of \(5,000\) feet. (b) Determine the altitude at which the temperature will be \(32^{\circ} \mathrm{F}\).

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Solve each of the problems algebraically. That is, set up an equation and solve it. Be sure to clearly label what the variable represents. Round your answer to the nearest tenth where necessary. The length of a rectangle is 12 more than four times its width. If the perimeter of the rectangle is 134 meters, find the dimensions for the rectangle.

Evaluate the given expression. $$|5-9|-|-3-8|$$

Perform the indicated operations and simplify as completely as possible. $$\left(2 x^{2}\right)\left(3 x^{3}\right)$$
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