Problem 64 - Algebra and Trigonometry

Question

Write each sentence as an inequality in two variables. Then graph the inequality. The y-variable is at least 2 more than the product of -3 and the x-variable.

Step-by-Step Solution

Answer

The inequality is \(y \geq -3x + 2\). To graph this, create a solid line for \(y = -3x + 2\) and shade the area above the line to represent all possible solutions to the inequality.

Step 1: Write the Inequality

The sentence 'The y-variable is at least 2 more than the product of -3 and the x-variable' can be translated into the inequality \(y \geq -3x + 2\). The wording 'at least' suggests greater than or equal to, while 'the product of -3 and the x-variable' can be written as '-3x' and '2 more' as '+2'.

Step 2: Graph the Inequality

To graph the inequality \(y \geq -3x + 2\), start by sketching the line \(y = -3x + 2\). This is a linear equation with a slope of -3 and a y-intercept of 2. Now, because this is a 'greater than or equal to' inequality, the points on the line are included in the solution. Therefore, the line should be solid. Finally, choose a test point not on the line (e.g., the origin, if it's not on the line) to determine which area to shade. If the test point satisfies the inequality, shade the area containing the test point. If not, shade the opposite side.

Key Concepts

Graphing Inequalities

Graphing inequalities is an essential skill in algebra that helps us visualize solutions on a coordinate plane. One key step is to first convert the inequality into an equation to find the boundary line. For example, we start with the inequality \(y \geq -3x + 2\) and initially sketch the line \(y = -3x + 2\), which serves as our boundary. This line behaves as a dividing line on the graph.
For the line like \(y = -3x + 2\), understand the importance of its characteristics:

• Slope: Indicates the steepness; here it's -3, meaning the line falls steeply from left to right.
• Y-Intercept: Where the line crosses the y-axis, which is at 2 in this case.

Since it's a 'greater than or equal to' inequality, every solution lies on or above this line, hence we use a solid line to showcase that boundary is included. After sketching the boundary line, a test point helps to determine which side of the line holds the solutions. Often, the origin (0, 0) is a reliable test point unless it's on the boundary line itself. By testing the origin in the inequality:

• If the inequality holds true with the test point, shade the region including the point.
• If not, shade the opposite region.

Incorporating these steps ensures a clear graph of the solution area for the inequality.

Two-Variable Inequalities

Two-variable inequalities involve expressions with two different variables, typically x and y, which broaden solutions from just numbers to entire regions on a graph. The central idea is to understand how these variables interact to form a relationship. In our case, the inequality \(y \geq -3x + 2\) describes a set of points that satisfy the condition of y being at least as much as \(-3x + 2\).
To better grasp this, think of:

• The y-variable: Represents the vertical positioning on the graph.
• The x-variable: Represents the horizontal positioning on the graph.

Two-variable inequalities allow us to visualize solutions that aren't isolated numbers. Instead, they form a collection of points setting a range of values for y depending on x. This opens up the possible combinations of x and y to satisfy the inequality, making the solution a "shaded region" above or below a boundary line, unlike equations that pinpoint specific intersections.
Therefore, the dynamic between x and y in two-variable inequalities helps in understanding the breadth of solutions, letting you see a detailed picture of how these relations play out on a graph.

Algebraic Expressions

Algebraic expressions form the foundation of effectively translating word problems into mathematical statements. In this process, terms like 'product', 'sum', 'increased by', and 'at least' are key indicators of the operations needed in forming expressions. For the original exercise, the sentence "The y-variable is at least 2 more than the product of -3 and the x-variable" becomes the algebraic expression \(-3x + 2\), combined with the inequality sign \(\geq\).
Breaking it down:

• Product of -3 and x: Indicates multiplication, giving us \(-3x\).
• '2 more than': Represents addition, leading to the expression \(-3x + 2\).
• 'Is at least': Translates to 'greater than or equal to', shown by \(\geq\).

Understanding these terms and operations is crucial, as it guides you from a verbal statement to a mathematical form that can be worked with efficiently. Algebraic expressions let you form equations and inequalities essential for solving problems in real-world and theoretical contexts. This bridge between language and math is where problems are first defined and let you apply algebraic rules to find solutions.