Problem 36 - Intermediate Algebra

Question

Graph each inequality in two variables. $$ 3 x-y \leq 4 $$

Step-by-Step Solution

Answer

Graph \(y = 3x - 4\) and shade above it.

Step 1: Rewrite Inequality in Slope-Intercept Form

First, let's rewrite the given inequality in a more familiar form: slope-intercept form, which is \(y = mx + b\). Start by isolating \(y\): \(-y \leq -3x + 4\)Now, multiply through by \(-1\) and flip the inequality sign:\(y \geq 3x - 4\).

Step 2: Identify the Boundary Line

The boundary line is given by the equation \(y = 3x - 4\). At this point, we treat the inequality as an equation to determine the line itself. This is a straight line with slope \(m = 3\) and y-intercept \(b = -4\).

Step 3: Graph the Boundary Line

Draw the boundary line on a graph. To plot \(y = 3x - 4\), start at the y-intercept (0, -4). Then use the slope (3) to find another point. From (0, -4), go up 3 units and right 1 unit to reach (1, -1). Connect these points with a solid line because the inequality includes 'equal to' (≥).

Step 4: Determine the Shaded Region

Since our inequality is \(y \geq 3x - 4\), shade the region above the line. This region represents all the (x, y) pairs that satisfy the inequality. To verify, pick a test point like (0, 0). Substitute into the inequality: \(0 \geq 3(0) - 4\), or \(0 \geq -4\), which is true, confirming the shading is correct.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a commonly used way to express the equation of a line. It is written as \( y = mx + b \), where:

• \( m \) is the slope, representing how steep the line is.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

To rewrite an inequality in slope-intercept form, you often start by isolating the \( y \) variable. For example, with the inequality \( 3x - y \leq 4 \), we isolate \( y \) by rearranging the terms:

1. Subtract \( 3x \) from both sides to get \(-y \leq -3x + 4\).
2. Then multiply every term by \(-1\), remembering to flip the inequality sign, resulting in \( y \geq 3x - 4 \).

This form makes it easier to graph the inequality, as you can directly identify the slope and the y-intercept.

Boundary Line

A boundary line is the line that represents the equation formed from an inequality. For example, for the inequality \( y \geq 3x - 4 \), the boundary line is given by the equation \( y = 3x - 4 \). To graph this line, start with the y-intercept. In this case, \( b = -4 \), so plot the point at (0, -4) on the graph. Next, use the slope \( m = 3 \), which indicates you rise 3 units and run 1 unit to the right. Thus, from (0, -4), move to (1, -1) by going up 3 and over 1. Draw a solid line through these points because the original inequality is inclusive (\( \geq \) or \( \leq \)). This line divides the graph into two distinct regions.

Shading Regions

Shading regions is a critical step when graphing inequalities. It demonstrates which side of the boundary line contains solutions to the inequality. When dealing with the inequality \( y \geq 3x - 4 \), the shaded region will be above the line.To determine which side to shade, choose a test point that isn't on the boundary line, such as (0, 0). Substitute this point into the inequality:

• \( 0 \geq 3(0) - 4 \)
• This simplifies to \( 0 \geq -4 \), which is a true statement.

Since the test point satisfies the inequality, you should shade the region of the graph that contains the point (0, 0). Always remember, if there are multiple parts of the graph, only the side where the inequality holds true throughout should be shaded.

Linear Inequalities

Linear inequalities are expressions that use inequality signs (\( >, <, \geq, \leq \)) instead of equal signs to compare two algebraic expressions. They are used to represent a range of possible values rather than a single fixed value.Unlike linear equations that result in a line on a graph, linear inequalities define areas in the coordinate plane. When graphing a linear inequality, like \( y \geq 3x - 4 \), first create the boundary line (as an equality), then determine which side of the line represents the solutions by shading that part.Key characteristics of linear inequalities include:

• The presence of a boundary line that is drawn solid for \( \geq \) or \( \leq \) and dashed for \( > \) or \( < \).
• A shaded region that signifies all possible solutions to the inequality.

Understanding these features helps you graphically depict inequalities, making it easier to visualize where all solution pairs \((x, y)\) lie.