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: Start by setting up the inequality y1 > y2. Substitute the given expressions for y1 and y2 into the inequality: \( \frac{2}{3}(6x - 9) + 4 > 5x + 1 \).
2. Step 2: Simplify the left-hand side of the inequality. Distribute \( \frac{2}{3} \) across \( (6x - 9) \): \( \frac{2}{3} \cdot 6x - \frac{2}{3} \cdot 9 + 4 > 5x + 1 \). This simplifies to \( 4x - 6 + 4 > 5x + 1 \).
3. Step 3: Combine like terms on the left-hand side: \( 4x - 2 > 5x + 1 \).
4. Step 4: Isolate x by subtracting \( 4x \) from both sides: \( -2 > x + 1 \). Then subtract 1 from both sides: \( -3 > x \), or equivalently \( x < -3 \).
5. Step 5: Represent the solution in interval notation. Since \( x < -3 \), the interval is \( (-\infty, -3) \).

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 values of y1 must be greater than those of y2 for certain values of x. Understanding how to manipulate and solve inequalities is crucial for determining the range of x that satisfies the given 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 of x, and analyzing their intersection points will help determine the values of x that satisfy the inequality.