Problem 44 - Algebra and Trigonometry

Question

In all exercises, other than \(\varnothing\). use interval notation to express solution sets and graph each solution set on a mumber line. In Exercises \(27-50\), solve each linear inequality. $$ 7-\frac{4}{5} x<\frac{3}{5} $$

Step-by-Step Solution

Answer

The solution to the inequality \(7-\frac{4}{5}x<\frac{3}{5} \) is \( x > 8 \). In interval notation, this is represented as \( (8,\infty) \).

Step 1: Simplify the inequality

Subtract 7 from both sides of the inequality: \(7-\frac{4}{5}x<\frac{3}{5} \Rightarrow -\frac{4}{5}x<\frac{3}{5} - 7 \Rightarrow -\frac{4}{5}x<-\frac{32}{5} \)

Step 2: Isolate the variable

Multiply both sides by \(-\frac{5}{4}\), and remember to flip the direction of the inequality as you are multiplying by a negative number: \(-\frac{4}{5}x \times -\frac{5}{4} > -\frac{32}{5} \times -\frac{5}{4} \Rightarrow x > 8\)

Step 3: Write in interval notation

The solution in interval notation would be \( (8,\infty) \) as \(x\) is greater than \(8\) and can go up to infinity.

Step 4: Graph on a number line

Draw a number line, mark the point \( 8 \) and because \(x\) is greater than \( 8 \) but not equal to \( 8 \) you will represent it by an open circle. Draw an arrow extending to the right of the number line, towards infinity.

Key Concepts

Understanding Interval Notation

Interval notation is a system used to describe a range of numbers by indicating the starting and ending points. This compact form replaces inequalities or sets with a clear, concise representation. For example, if you have a number greater than 8 that extends indefinitely, you would express this in interval notation as \( (8, \infty) \).

In interval notation, different brackets serve specific purposes:

• Parentheses \( ( or ) \) denote that the endpoint is not included in the set, which is also known as an open interval.
• Brackets \[ [ or ] \] imply that the endpoint is included, known as a closed interval.

A number greater than 8 but less than 10 would be \( (8, 10) \), while including the 10 would be \( (8, 10] \). Understanding how to read and write interval notation is crucial for communicating solution sets clearly and effectively.

Deciphering Inequality Solution Sets

Inequality solution sets define the range of values that satisfy an inequality. In our exercise, the inequality \( 7 - \frac{4}{5} x < \frac{3}{5} \) describes a set of values for \(x\) that make the inequality true. After isolating \(x\) and finding \( x > 8 \) in the steps provided, the solution set includes all numbers greater than 8.

To portray solution sets:

• Use interval notation, where our example translates to \((8, \infty)\).
• Note the distinction between \(>\) and \(\ge\) or \(<\) and \(\le\) to determine if the endpoint is included or not.
• In this case, since it's \(>8\) rather than \(\ge8\), 8 is not part of the solution set, thus the parenthesis rather than a bracket.

Grasping these details helps in understanding the full scope of solutions without ambiguity.

Graphing Solution Sets on a Number Line

Graphing on a number line visually represents the solution set of an inequality, aiding in comprehension and verification. After finding the solution \( x > 8 \) for the inequality given, we can graph this on a number line:

• First, draw a straight horizontal line and mark the point at 8.
• Since 8 is not included in the solution set, we place an open circle at 8 to indicate that 8 itself is not a solution.
• An arrow extending from the open circle towards higher values represents all numbers greater than 8.

Visualizing the solution this way confirms that numbers to the right of 8, moving towards infinity, are all valid solutions, emphasizing the continuous nature of the set of numbers involved.