Problem 37 - Introductory Algebra for College Students

Question

Write each sentence as a linear inequality in two variables. Then graph the inequality. The sum of the \(x\) -variable and the \(y\) -variable is at least 2

Step-by-Step Solution

Answer

\(x + y \geq 2\) is the linear inequality and the region above and including the line \(x+y = 2\) on the graph represents all the possible solutions.

Step 1: Translate Sentence to Inequality

Given the sentence 'The sum of the x -variable and the y -variable is at least 2', you apply the knowledge that 'the sum of' represents addition and 'is at least' corresponds to 'greater than or equal to' in mathematical terms. After applying this knowledge, the sentence can be translated into the inequality \(x + y \geq 2\).

Step 2: Graph the Inequality

First, graph the line correspond to the equality \(x+y = 2\) which is a straight line that passes through the points (0,2) and (2,0). Because of the greater than or equal sign, the valid solution to the inequality will be the area above this line or on the line itself. Hence shade the region above and including the line \(x+y = 2\) to represent all the possible solutions for this inequality.

Key Concepts

Graphing Inequalities

When graphing linear inequalities, we start by understanding the equality version of the inequality. This means we replace the inequality symbol with an equals sign and graph the resulting line.
In our example, the inequality is \(x + y \geq 2\). We first graph the line \(x + y = 2\), which involves finding two points that solve this equation. The points (0, 2) and (2, 0) are solutions, so plotting these points and drawing a line through them gives the boundary of our inequality.
The type of line drawn is also crucial:

• If the inequality is \(\geq\) or \(\leq\), like in our case, draw a solid line to show that points on the line are included in the solution.
• If the inequality is \(>\) or \(<\), use a dashed line to indicate the line itself is not part of the solution.

After the line is drawn, shade the appropriate side that satisfies the inequality. For \(x + y \geq 2\), shade the region above the line, as this represents values of \(x\) and \(y\) whose sum is 2 or greater.

Two-variable Inequalities

Two-variable inequalities involve two variables, usually denoted as \(x\) and \(y\). These types of inequalities define a region on the coordinate plane that contains all solutions to the inequality.
To solve, follow these steps:

• Transform the inequality into its equation form to find the boundary line.
• Determine which side of the boundary contains the solutions by testing points.
• Shade the region that satisfies the inequality. If the test point satisfies the inequality, shade that side of the line.

The region that is shaded represents all pairs of \((x, y)\) that satisfy the inequality. In the context of the example \(x + y \geq 2\), all possible solutions lie on or above the line that runs diagonally across the plane from (0,2) to (2,0). This represents every combination of \((x, y)\) where their sum is at least 2.

Mathematical Translation

Understanding how to translate statements into mathematical inequalities is an essential skill in math. It involves interpreting phrases from natural language into algebraic expressions or inequalities.
When given a sentence or phrase:

• Identify the mathematical operations indicated by words such as "sum," "difference," "product," and "quotient."
• Translate words like "at least," "at most," "no more than," and "less than" into inequality symbols (\(\geq\), \(\leq\), \(<\), \(>\)).

For instance, in the exercise, the phrase "the sum of the \(x\)-variable and the \(y\)-variable is at least 2" was translated to the inequality \(x + y \geq 2\). Remember, translation is not only about changing words to symbols but ensuring the symbols properly reflect the meaning in context of the problem. This skill allows students to convert real-world problems into solvable mathematical expressions or inequalities.