Problem 26 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ x-8>5 x+3 $$

Step-by-Step Solution

Answer

The solution is \( x < -\frac{11}{4} \) or \((-\infty, -\frac{11}{4})\).

Step 1: Identify the Inequality

The given inequality is \( x - 8 > 5x + 3 \). Our goal is to find the values of \( x \) that satisfy this inequality.

Step 2: Move All Terms Involving x to One Side

Subtract \( x \) from both sides of the inequality to get:\[ -8 > 4x + 3 \]

Step 3: Move Constant Terms to the Other Side

Subtract 3 from both sides of the inequality:\[ -8 - 3 > 4x \]Simplify the left side to get:\[ -11 > 4x \]

Step 4: Isolate x

Divide both sides by 4 to solve for \( x \). Since 4 is positive, the inequality sign remains the same:\[ x < -\frac{11}{4} \]

Step 5: Express the Solution in Interval Notation

The solutions for \( x \) that satisfy the inequality are all numbers less than \(-\frac{11}{4}\). In interval notation, this is written as:\[ (-\infty, -\frac{11}{4}) \]

Key Concepts

Interval Notation

Interval notation is a powerful tool used to describe a set of numbers between two endpoints, which helps in expressing solutions of inequalities concisely. In our particular exercise, once we solved the inequality to find the solution of \( x < -\frac{11}{4} \), we transitioned this into the interval notation as \((-\infty, -\frac{11}{4})\).
This interval tells us that the solutions include all numbers starting from negative infinity up to, but not including, \(-\frac{11}{4}\).

• Parentheses \(()\) are used when the endpoint is not part of the solution (known as an open interval).
• Brackets \([]\) are used when the endpoint is included (known as a closed interval), though our solution does not include the \( -\frac{11}{4} \), so we use a parenthesis.

This notation provides a simple way to represent infinite or finite ranges of numbers, facilitating easy communication of the solution set.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifing equations or inequalities to isolate the variable of interest, in this case, \( x \).
In our exercise, we began with the inequality: - Step 1: \( x - 8 > 5x + 3 \)Our goal was to isolate \( x \). First, we moved all terms involving \( x \) to one side: - Subtract \( x \) from both sides, resulting in \( -8 > 4x + 3 \).
- Then, we move constant terms to the opposite side: - Subtract 3 from both sides, leading to \( -11 > 4x \).Finally, we isolated \( x \) by dividing both sides by 4.
This step required careful consideration, especially with regard to maintaining the inequality sign correctly, as we divided by a positive number, thus retaining \( < \), resulting in \( x < -\frac{11}{4} \).
Algebraic manipulation is crucial for solving inequalities as it simplifies complex expressions into manageable forms.

Inequality Properties

Understanding inequality properties is essential when solving inequalities, as they determine how operations affect the direction or expression of the relationship between numbers.
In our solution:- The crucial property is maintaining the direction of the inequality.When subtracting the same number from both sides, such as moving \( x \) and the constant 3, the inequality direction remains intact. Similarly, when dividing or multiplying by a positive number, the inequality sign does not change.
However, it's vital to remember that if you multiply or divide by a negative number, the inequality sign reverses direction, though this did not occur in our exercise.
Another key concept is recognizing the solution as a range rather than a distinct number, as demonstrated by our interval \( x < -\frac{11}{4} \).
This emphasizes the importance of inequality properties in preserving the correct relationships and accurately expressing solutions.