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Chapter 2: Problem 69

Write an equivalent inequality. All real numbers strictly between -6 and 6 .

Short Answer

Expert verified
The inequality is \(-6 < x < 6\).

Step by step solution

01

Understanding the Range

We are given that the solution set includes all real numbers that are strictly between -6 and 6. This means that the solution set does not include -6 or 6 themselves.
02

Writing the Inequality

To express all real numbers strictly between -6 and 6, we use the following inequality: \(-6 < x < 6\). This inequality means that x is greater than -6 and less than 6.

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Key Concepts

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

Real Numbers
When we talk about real numbers, we are referring to a wide set of numbers that include several subsets. This means all possible numbers that can be plotted on a number line.
Some of the main categories within real numbers are:
  • Integers: Whole numbers that can be positive, negative, or zero. For instance, -3, 0, and 4 are integers.
  • Rational numbers: Numbers that can be expressed as a fraction of two integers. For example, 1/2 and -3/4 are rational numbers.
  • Irrational numbers: Numbers that cannot be expressed as a fraction. These include numbers like \( \pi \) or the square root of 2.
When dealing with inequalities and solution sets, we often examine how a group of these real numbers relate to each other in terms of greater than, less than, or equal scenarios. Understanding real numbers is foundational for interpreting and writing inequalities.
Solution Set
A solution set represents all the values that satisfy a given inequality or equation. In simpler terms, it is the range of numbers that make the inequality true.
In the context of our exercise, the solution set is all real numbers that are strictly between -6 and 6. It does not include -6 or 6 themselves. Inequalities often use open intervals to represent these conditions:
  • Open interval: The interval \(-6 < x < 6\) is open, meaning it includes every number greater than -6 but less than 6. The endpoints -6 and 6 are not included.
  • Closed interval: Although it is not the case in this exercise, closed intervals (like \[a, b\]) include the endpoints. So this would mean both \(a\) and \(b\) are part of the solution set.
Being able to clearly define a solution set helps you to properly express and understand which numbers complete the inequality's requirements.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations (such as addition, subtraction, multiplication, and division) that represent a specific value or relationship.
In our inequality exercise, the algebraic expression is \(-6 < x < 6\), which incorporates a variable \(x\) in relation to constant real numbers -6 and 6. Here's how they function:
  • Expressions as inequalities: In inequalities, these algebraic expressions show a range instead of a specific numerical result. The inequality indicates the relationship between variables and numbers.
  • Manipulating expressions: Understanding and altering these expressions is crucial in solving problems, such as isolating variables or converting inequalities to equivalent forms.
Interpreting algebraic expressions accurately in terms of inequalities helps solve problems and understand mathematical relationships better.

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Most popular questions from this chapter

Set up a compound inequality for the following and then solve. The perimeter of a square must be between 40 feet and 200 feet. Find the length of all possible sides that satisfy this condition.

Set up a compound inequality for the following and then solve. A certain antifreeze is effective for a temperature range of \(-35^{\circ} \mathrm{C}\) to \(120^{\circ} \mathrm{C}\). Find the equivalent range in degrees Fahrenheit.

An equilateral triangle with sides measuring 10 units is similar to another equilateral triangle with scale factor of 2: 3 . Find the perimeter of the unknown triangle.

Graph all solutions on a number line and provide the corresponding interval notation. $$ x>-2 $$

If triangle \(A B C\) is similar to triangle RST, then find the remaining two sides given the following. \(a=4, b=9, c=12,\) and \(s=3\)
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