Use interval notation to represent all values of x satisfying the given conditions. y = 2x - 11 + 3(x + 2) and y is at most 0

Verified step by step guidance

1. Step 1: Start by simplifying the given equation y = 2x - 11 + 3(x + 2). Combine like terms to simplify the expression for y. Distribute the 3 across (x + 2) to get y = 2x - 11 + 3x + 6.
2. Step 2: Combine the terms involving x and the constants. This results in y = (2x + 3x) + (-11 + 6), which simplifies to y = 5x - 5.
3. Step 3: The problem states that y is at most 0, which means y ≤ 0. Substitute the simplified expression for y into the inequality: 5x - 5 ≤ 0.
4. Step 4: Solve the inequality 5x - 5 ≤ 0. First, isolate the term involving x by adding 5 to both sides: 5x ≤ 5.
5. Step 5: Divide both sides of the inequality by 5 to solve for x: x ≤ 1. Represent this solution in interval notation as (-∞, 1].

Key Concepts

Linear Equations

A linear equation is an algebraic expression that represents a straight line when graphed. It typically takes the form y = mx + b, where m is the slope and b is the y-intercept. In this question, the equation y = 2x - 11 + 3(x + 2) must be simplified to identify the values of x that satisfy the condition y ≤ 0.

Inequalities

An inequality is a mathematical statement that compares two expressions, indicating that one is less than, greater than, or equal to the other. In this case, we are interested in the inequality y ≤ 0, which means we need to find the range of x values for which the linear expression results in a value of y that is less than or equal to zero.

Interval Notation

Interval notation is a way of representing a range of values on the number line. It uses brackets and parentheses to indicate whether endpoints are included or excluded. For example, [a, b] includes both a and b, while (a, b) excludes them. In this problem, once the values of x that satisfy the inequality are determined, they will be expressed in interval notation to clearly communicate the solution set.