Problem 69 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Key Concepts

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.