Problem 32 - College Algebra with Modeling and Visualization

Question

Solve the inequality symbolically. Express the solution set in set-builder or interval notation. $$ -4 \leq 5-\frac{4}{5} x<6 $$

Step-by-Step Solution

Answer

The solution is \(( -\frac{5}{4}, \frac{45}{4}]\).

Step 1: Split the Compound Inequality

The compound inequality is: \[-4 \leq 5 - \frac{4}{5}x < 6\].To solve it, we split this into two separate inequalities: 1. \(-4 \leq 5 - \frac{4}{5}x\)2. \(5 - \frac{4}{5}x < 6\).

Step 2: Solve the First Inequality

For the inequality \(-4 \leq 5 - \frac{4}{5}x\):1. Subtract 5 from both sides to isolate the term with \(x\): \[-9 \leq -\frac{4}{5}x\]2. Multiply both sides by \(-\frac{5}{4}\) to solve for \(x\). Remember to flip the inequality sign: \[x \leq \frac{45}{4}\].

Step 3: Solve the Second Inequality

For the inequality \(5 - \frac{4}{5}x < 6\):1. Subtract 5 from both sides: \[-\frac{4}{5}x < 1\]2. Multiply both sides by \(-\frac{5}{4}\). Remember to flip the inequality sign: \[x > -\frac{5}{4}\].

Step 4: Combine the Solutions

Combining the results from Step 2 and Step 3 gives us:\[-\frac{5}{4} < x \leq \frac{45}{4}\].

Step 5: Express the Solution in Interval Notation

The solution set for \(-\frac{5}{4} < x \leq \frac{45}{4}\) in interval notation is:\(( -\frac{5}{4}, \frac{45}{4}]\).

Key Concepts

Compound Inequalities

A compound inequality consists of two separate inequalities that are combined into one statement. In mathematics, it's used to express a range of values that satisfy more than one condition.
For instance, the compound inequality \[-4 \leq 5 - \frac{4}{5}x < 6\]breaks down into two separate parts:

• \(-4 \leq 5 - \frac{4}{5}x\)
• \(5 - \frac{4}{5}x < 6\)

Each part needs to be true for the compound inequality to hold. It's similar to having two separate inequality statements that describe either a more relaxed or a more constrained condition than single inequalities. Thus, compound inequalities work to refine the boundary conditions for solutions in arithmetic or algebra.

Inequality Solving

Solving inequalities is not too different from solving equations, but with a few key distinctions. When you solve an inequality, you endeavor to find the set of values (or range) for the variable that makes the inequality true.
To solve inequalities:

• Isolate the variable on one side of the inequality.
• Perform similar operations as you would with equations (addition, subtraction, multiplication, or division).
• Be careful: multiplying or dividing by a negative number flips the inequality sign.

For our exercise, the focus was on splitting the compound inequality into separate ones, solving each, and remembering that crucial step - whenever you multiply or divide by a negative number, always flip the direction of the inequality.

Interval Notation

Interval notation is a shorthand method of writing the solution set of inequalities. It provides a clear visual representation of the solution set using parentheses and brackets.
Parentheses \(()\) indicate that a particular number is not included, while brackets \([]\) signify inclusion. Consider the solution \(-\frac{5}{4} < x \leq \frac{45}{4}\) in interval notation. This is represented as \(( -\frac{5}{4}, \frac{45}{4}]\).

• The parenthesis \(()\) before \(-\frac{5}{4}\) suggests \(x\) cannot be \(-\frac{5}{4}\).
• The bracket \([]\) after \(\frac{45}{4}\) shows \(\frac{45}{4}\) is a permissible solution.

Thus, interval notation efficiently communicates a range of potential values, perfect for describing solutions of inequalities.

Set-Builder Notation

Set-builder notation is another method for describing sets. It is particularly helpful in mathematics to succinctly convey the conditions an element must meet to be included in the set.
Typically, set-builder notation uses curly braces \({}\) to define limits on solutions. In this notation, the solution to the exercise is written as\(\{ x\,|\, -\frac{5}{4} < x \leq \frac{45}{4} \}\).

• This reads as "the set of all numbers \(x\) such that \(-\frac{5}{4} < x\) and \(x \leq \frac{45}{4}\)."
• The vertical bar \(|\) is read as "such that."

This notation is especially flexible, as it allows for the presentation of complex numbers and conditions within simple brackets, ensuring clarity when representing more complicated solution sets.