Problem 72 - Algebra for College Students

Question

Give, in interval notation, the unknown numbers in each description. Three times a number, minus \(5,\) is no more than 7 .

Step-by-Step Solution

Answer

Interval notation: \((-\infty, 4]\)

Step 1 - Set Up the Inequality

Translate the verbal description into a mathematical inequality. Let the unknown number be represented by the variable \(x\). The description 'three times a number, minus 5, is no more than 7' can be written as: \[3x - 5 \leq 7 \]

Step 2 - Isolate the Variable

To isolate \(x\), first add \(5\) to both sides of the inequality: \[3x - 5 + 5 \leq 7 + 5 \rightarrow 3x \leq 12 \]

Step 3 - Solve for x

Divide both sides of the inequality by \(3\) to solve for \(x\): \[ \frac{3x}{3} \leq \frac{12}{3} \rightarrow x \leq 4 \]

Step 4 - Express the Solution in Interval Notation

The solution to the inequality can now be expressed in interval notation. Since \(x\) is less than or equal to \(4\), the interval is: \[(-\infty, 4] \]

Key Concepts

interval notation

Interval notation is a shorthand way of expressing ranges of numbers. It is especially useful when describing the solutions to inequalities.
In interval notation, a pair of numbers is used to represent the set of all numbers between them.
There are two types of brackets used in interval notation:

• Square brackets [ ] indicate that the endpoint is included in the interval.
• Parentheses ( ) indicate that the endpoint is not included in the interval.

For example:
– The interval \( [1, 5] \) represents all numbers between 1 and 5, including 1 and 5.
– The interval \( (2, \infty) \) represents all numbers greater than 2, but not including 2.
In our exercise, the inequality \( x \leq 4 \) is expressed in interval notation as \( (-\infty, 4] \). This means that the solution includes all values less than or equal to 4.

mathematical inequality

Mathematical inequalities are used to show that one side of an equation is greater than, less than, or equal to the other side.
Common symbols in inequalities include:

• \(< \): Less than
• \(> \): Greater than
• \(\leq \): Less than or equal to
• \(\geq \): Greater than or equal to

Inequalities are a way to compare expressions and figure out the range of possible values for a variable.
For example, in the exercise, we started with the inequality \((3x - 5 \leq 7) \). This inequality shows that three times a number, minus 5, is no more than 7.
Understanding and manipulating inequalities is crucial for solving a wide range of mathematical problems.

isolate variables

Isolating variables is the process of solving an equation or inequality for one specific variable.
This is done by performing operations that move other terms to the opposite side of the equation or inequality.
Here are the steps to isolate a variable:

• Identify the variable you need to isolate.
• Use addition or subtraction to remove constants from the variable's side.
• Use multiplication or division to remove coefficients from the variable.

In the given exercise, we started with the inequality \(3x - 5 \leq 7 \):

• First, we added 5 to both sides to get \(3x \leq 12 \).
• Then, we divided both sides by 3 to get \( x \leq 4 \).

By following these steps, we isolated the variable \( x \) and solved the inequality.
Isolating variables is a key skill for solving equations and inequalities.