Problem 16 - A Graphical Approach to Precalculus with Limits

Question

Graph each inequality. $$x<3+2 y$$

Step-by-Step Solution

Answer

Graph the line \(y = \frac{1}{2}x - \frac{3}{2}\) as dashed, and shade above it.

Step 1: Rearrange the Inequality

First, we need to rearrange the inequality to identify the slope and y-intercept for graphing. The initial inequality is \(x < 3 + 2y\). We need to solve for \(y\) to make it easier to graph. Begin by subtracting 3 from both sides to get \(x - 3 < 2y\).

Step 2: Solve for y

Now divide every term in the inequality \(x - 3 < 2y\) by 2 to solve for \(y\). This gives us \(\frac{x}{2} - \frac{3}{2} < y\). To convert it to the standard form of \(y > \ldots\), we can write it as \(y > \frac{1}{2}x - \frac{3}{2}\).

Step 3: Graph the Associated Equality

To graph the inequality \(y > \frac{1}{2}x - \frac{3}{2}\), first graph the line \(y = \frac{1}{2}x - \frac{3}{2}\) as a dashed line. The dashed line indicates that points on the line are not included in the solution of the inequality.

Step 4: Determine the Correct Region

After drawing the dashed line, determine which region \(y > \frac{1}{2}x - \frac{3}{2}\) covers. Choose a test point not on the line, such as (0,0). Substitute into \(y > \frac{1}{2}x - \frac{3}{2}\): \(0 > \frac{1}{2}(0) - \frac{3}{2}\) simplifies to \(0 > -\frac{3}{2}\), which is true. Thus, the region above the line is shaded as part of the solution.

Step 5: Finalize the Graph

Shade the entire region above the dashed line on the graph. This represents all the solutions \((x, y)\) pairs that satisfy \(y > \frac{1}{2}x - \frac{3}{2}\). Remember, the line itself is not included, as indicated by the line being dashed.

Key Concepts

Rearranging Inequalities

Rearranging inequalities is a crucial step when solving or graphing them. Often, inequalities are not given in a form that is easy to interpret or graph right away. To graph an inequality, it's helpful to rearrange it to solve for one variable in terms of the other. Consider the inequality \(x < 3 + 2y\). To make graphing easier, we need to express it in terms of \(y\). This means isolating \(y\) on one side of the inequality.

• Start by subtracting 3 from both sides: \(x - 3 < 2y\).
• Next, divide every term by 2: \(\frac{x}{2} - \frac{3}{2} < y\).
• Rearrange it to the more common form \(y > \frac{1}{2}x - \frac{3}{2}\), where the terms are clear for graphing.

This process not only helps in identifying the slope and y-intercept but also changes the direction of the inequality which is essential for the graphical representation.

Slope-Intercept Form

To effectively graph inequalities, you should become familiar with the slope-intercept form of a line, which is typically written as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

• In our rearranged inequality \(y > \frac{1}{2}x - \frac{3}{2}\), the coefficient \(\frac{1}{2}\) represents the slope \(m\).
• The constant \(-\frac{3}{2}\) is the y-intercept \(b\), indicating where the line crosses the y-axis.

Understanding this form helps you quickly plot the slope and y-intercept on a graph to draw the line. The slope indicates the line's steepness, while the y-intercept tells you where the line touches the y-axis. These components are foundational to accurately depicting both the equality (line) and inequality (region) on a graph.

Shading Regions

When graphing inequalities, shading regions is about representing all the solutions that satisfy the inequality. Once you have drawn the line associated with the equation, you need to determine on which side of the line to shade. For \(y > \frac{1}{2}x - \frac{3}{2}\), the line \(y = \frac{1}{2}x - \frac{3}{2}\) is the boundary.

• The dashed line indicates that points on the line are not included in the solution set. This is because the inequality is strictly greater than (\(>\)), not greater than or equal to (\(\geq\)).
• Once the boundary is established, shading the region above the line represents all points \((x, y)\) where the inequality holds true.

This visual representation helps in understanding the range of solutions by clearly marking all valid \((x, y)\) pairs that satisfy the inequality.

Testing Points in Inequalities

Testing points is an effective technique to determine which region in the graph is the solution for an inequality. After plotting the boundary line, you need to decide which side satisfies the inequality. A common strategy is to choose a test point.

• Select a point that is not on the line, such as \((0,0)\), for its simplicity.
• Substitute this point into the inequality. For \(y > \frac{1}{2}x - \frac{3}{2}\), substitute \((0,0)\): \(0 > \frac{1}{2}(0) - \frac{3}{2}\).
• Simplify to see if the inequality holds: \(0 > -\frac{3}{2}\), which is true.

Thus, the region containing \((0,0)\), which is above the line, is the correct shaded region. Testing points confirms our shading decision, ensuring that all selected solutions satisfy the inequality.