Problem 107 - College Algebra Essentials

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When graphing \(3 x-4 y<12,\) it's not necessary for me to graph the linear equation \(3 x-4 y=12\) because the inequality contains a \(<\) symbol, in which equality is not included.

Step-by-Step Solution

Answer

The statement does not make sense, because even for inequalities that do not include equality, the related equation is graphed, but with a dashed line instead of a solid one.

Step 1: Understand the statement

The statement suggests that it's not necessary to graph the equation \(3x - 4y = 12\) while graphing the inequality \(3x - 4y < 12\) as the inequality does not include equality.

Step 2: Analyze the Graphing Process for Inequalities

While graphing inequalities, we usually first graph the related equation (which is the boundary line). The difference is in the type of line we use. For inequalities that include equality (<= or >=), the line is solid indicating that points on the line are solutions. For inequalities that do not include equality (< or >), the line is dashed, meaning points on the line do not satisfy the inequality.

Step 3: Determine the Validity of the Statement

As per the graphing process described in step 2, even when graphing inequalities that do not include equality, we still graph the related equation. The difference is that a dashed line is used instead of a solid one. Therefore, the statement does not make sense.

Key Concepts

Graphing Inequalities

When you're graphing inequalities like \(3x - 4y < 12\), you're essentially working with a boundary that separates the solutions from the non-solutions. Inequalities help us determine which region of the graph represents the set of possible solutions. This is done by plotting the "boundary line," which is derived from the equation by replacing the inequality symbol with an equals sign. For instance, in \(3x - 4y < 12\), the boundary line is \(3x - 4y = 12\).

• The region above or below the boundary line is filled based on the inequality's requirements.
• This filled region is where all solutions to the inequality are found.
• The boundary itself might not be included in the solution set, depending on whether it’s represented by a dashed line or a solid line.

It's essential to graph the boundary line first, as it provides a visual representation to test which region satisfies the inequality.

Boundary Line

The boundary line is a crucial part of graphing inequalities. It's the line you draw on the graph, which acts as a divide between solutions and non-solutions. For our example \(3x - 4y = 12\), this line serves as the tipping point:

• To create the boundary line, set the inequality to an equality, replacing '<' or '>' with '='.
• You then plot this line on the coordinate plane using techniques similar to graphing linear equations.
• This line separates the coordinate plane into two distinct regions.

Once the line is drawn, you'll consider the type of line it should be — dashed or solid, based on the inequality symbol. The boundary line itself may or may not be part of the solution, which we'll convey using different line styles.

Dashed Line

When graphing inequalities that use '<' or '>', such as \(3x - 4y < 12\), draw the boundary as a dashed line. This dashed line indicates that the values on the boundary line do not satisfy the inequality. In other words, any point you choose on this line will not make \(3x - 4y < 12\) true.

• A dashed line tells us that the equality is not part of the solution set.
• It visually implies that while values nearby can be considered solutions, those precisely on the line are not.
• This approach aids in quickly determining which region of the graph satisfies the inequality after the line is plotted.

Therefore, remember to use a dashed line whenever you work with '<' or '>' in your inequalities to correctly represent the boundary.

Solid Line

If your inequality includes '≤' or '≥', like \(3x - 4y \leq 12\), then the boundary line should be solid. A solid line signifies that points on the line satisfy the inequality. It's as if the line itself is part of the solution.

• A solid line visually shows inclusion, meaning any point on this boundary fills the inequality.
• This inclusion contrasts with a dashed line, where equality is not part of the solution.
• Using a solid line helps indicate that solutions exist on the line itself, as well as in the adjacent region that meets the inequality’s condition.

Choosing the correct type of line – dashed or solid – is vital in accurately graphing inequalities and determining which portion of the plane satisfies the given conditions.