In Exercises 95–102, use interval notation to represent all values of x satisfying the given conditions. y1 = (2/3)(6x - 9) + 4, y2 = 5x + 1, and y1 > y2

Verified step by step guidance

1. <Step 1: Set up the inequality.> Start by setting up the inequality based on the given conditions: \( y_1 > y_2 \). This translates to \( \frac{2}{3}(6x - 9) + 4 > 5x + 1 \).
2. <Step 2: Simplify the left side of the inequality.> Distribute \( \frac{2}{3} \) across the terms inside the parentheses: \( \frac{2}{3} \times 6x - \frac{2}{3} \times 9 \). This simplifies to \( 4x - 6 \). So, the inequality becomes \( 4x - 6 + 4 > 5x + 1 \).
3. <Step 3: Combine like terms.> Simplify the left side by combining like terms: \( 4x - 2 > 5x + 1 \).
4. <Step 4: Isolate the variable x.> Subtract \( 4x \) from both sides to get \( -2 > x + 1 \). Then, subtract 1 from both sides to isolate \( x \): \( -3 > x \).
5. <Step 5: Write the solution in interval notation.> The inequality \( -3 > x \) can be expressed in interval notation as \( (-\infty, -3) \). This represents all values of \( x \) that are less than \( -3 \).

Key Concepts

Interval Notation

Interval notation is a mathematical notation used to represent a range of values on the number line. It uses parentheses and brackets to indicate whether endpoints are included (closed intervals) or excluded (open intervals). For example, the interval (2, 5] includes all numbers greater than 2 and up to 5, including 5 but not 2.

Inequalities

Inequalities express the relationship between two expressions that are not necessarily equal. In this context, the inequality y1 > y2 indicates that the value of the function y1 must be greater than that of y2 for certain values of x. Solving inequalities often involves finding the range of x values that satisfy the condition.

Linear Functions

Linear functions are mathematical expressions that create a straight line when graphed. They can be represented in the form y = mx + b, where m is the slope and b is the y-intercept. In this problem, both y1 and y2 are linear functions, and understanding their slopes and intercepts is crucial for determining where one function is greater than the other.