Problem 67 - Algebra for College Students

Question

Give, in interval notation, the unknown numbers in each description. Six times a number is between -12 and 12 .

Step-by-Step Solution

Answer

(-2, 2).

Step 1: Write the Inequality

Six times a number is represented as \(6x\). According to the problem, \(6x\) is between -12 and 12. Therefore, we can write the compound inequality as: -12 < 6x < 12.

Step 2: Isolate the Variable

Simplify the fractions: -2 < x < 2.

Step 3: Write the Solution in Interval Notation

The solution \(-2 < x < 2\) in interval notation is written as (-2, 2).

Key Concepts

Compound Inequalities

Compound inequalities involve two separate inequalities that are combined into one statement.
In this exercise, the number multiplied by six lies between -12 and 12.
We write this as a compound inequality: \(-12 < 6x < 12\).
This means the value of \(6x\) is greater than -12 and less than 12 at the same time.
Understanding how to read and write compound inequalities is essential when working with ranges of values.

Isolate the Variable

To solve the compound inequality, we need to isolate the variable \(x\).
This means we need to get \(x\) alone on one side of the inequality.
First, we have: \(-12 < 6x < 12\).
We can simplify this by dividing all sides of the inequality by 6.
This gives us: \(\frac{-12}{6} < x < \frac{12}{6}\), which simplifies to: \(-2 < x < 2\).
At this point, \(x\) is isolated, and we have a clearer understanding of its possible values.

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operators.
In our example, \(6x\) is an algebraic expression where \(6\) is the coefficient and \(x\) is the variable.
When dealing with algebraic expressions, especially in inequalities, understanding the structure of the expression helps in manipulating it effectively.
This is crucial when converting between forms, such as moving from the equation to interval notation.

Inequalities Solutions

Once we have the simplified form of the compound inequality, \(-2 < x < 2\), the next step is to express this solution in interval notation.
Interval notation is a way of writing subsets of the real number line and is very useful for representing ranges of solutions.
Here, \(-2 < x < 2\) translates to \(-2, 2\) in interval notation.
This indicates that \(x\) can take any value between -2 and 2, but not including -2 and 2 themselves.
Understanding interval notation is an important skill for visualizing and communicating the solution set of an inequality.