Problem 64 - College Algebra

Question

In Exercises 63–64, 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 \(y \geq -3x+2\). The graph consists of the line \(y = -3x + 2\) and the shaded region is everything above this line because that is where \(y\) is greater than -3 times \(x\) plus 2.

Step 1 - Translating the sentence into a mathematical inequality

The sentence 'The \(y\)-variable is at least 2 more than the product of -3 and the \(x\)-variable' means that \(y\) is greater than or equal to -3 times \(x\) plus 2. This can be written as the inequality: \(y \geq -3x+2\).

Step 2 - Graphing the inequality

To graph this inequality, start with graphing the line \(y = -3x + 2\). This line divides the plane into two regions. Using a test point, you can determine which region is the solution of the inequality.

Step 3 - Choosing a test point

Select a test point not on the line. The origin (0,0) is a good option here. Substituting these coordinates into the original inequality, you get \(0 \geq -3*0 + 2\) which simplifies to \(0 \geq 2\). This is false, therefore (0,0) lies in the non-solution half-plane.

Step 4 - Shading the solution half-plane

Since (0,0) is in the non-solution half-plane, the other half of the plane is the solution. Shade the half-plane not containing the origin, and that's your valid solution region.

Key Concepts

Graphing Inequalities

Graphing an inequality involves drawing the boundary line on a coordinate plane, and then determining which side of the line contains the solutions to the inequality. Here's how you do it in a few simple steps:

• First, understand the relationship as an equation. For example, starting with an equation like \(y = -3x + 2\) helps you know where to draw the line.


• Use a test point, like \((0,0)\), to decide which side of the line the solutions lie on. This involves checking if the test point satisfies the inequality.


• Finally, shade the region that represents all possible solutions to the inequality. If the test point does not satisfy the inequality, shade the opposite side of the line.

It's super important to choose the correct test point and to understand the relationship of the inequality with the boundary line (dotted for \(>\) or \(<\), solid for \(\geq\) or \(\leq\)). This ensures precision in showcasing all potential solutions on a graph.

Linear Inequalities

Linear inequalities in two variables, such as \(y \geq -3x + 2\), are inequalities that graph as a line or boundary which partitions the plane into two regions. Here's what you need to grasp:

• This type of inequality involves linear expressions in terms of two variables \(x\) and \(y\).


• A linear inequality does not just constrain \(y\) or \(x\) alone but places a combined constraint on both.


• The boundary line derived from the equivalent equation, like \(y = -3x + 2\), is crucial as it acts as a starting point for shading the area representing solutions.

The boundary line in a graph, depending on the inequality symbol, might be solid (inclusive) or dashed (exclusive). Understanding these symbols is critical for interpreting the graph correctly. Linear inequalities typically mean finding a region on the graph that fulfills a given condition instead of just points.

Two-variable Inequalities

Two-variable inequalities incorporate both \(x\) and \(y\) to describe a region of solutions on a graph. Understanding these inequalities involves recognizing a few important points:

• Each solution to the inequation has two components: an \(x\) value and a corresponding \(y\) value.


• Typically found in real-world situations where one factor depends on another, like expenses depending on the quantity bought.


• The solutions are not just a single set of values but an infinite set of possible \((x, y)\) pairs that create a shaded region when plotted.

To truly master two-variable inequalities, it's crucial to practice plotting and interpreting the solutions on graphs, which shows all possible scenarios meeting the inequality's requirement. This skill informs better decision-making in applied contexts.