Problem 46 - A Graphical Approach to College Algebra

Question

Graph the solution set of each system of inequalities by hand. $$\begin{array}{r} x-y < 1 \\ -1 < y < 1 \end{array}$$

Step-by-Step Solution

Answer

The solution set is the region directly below the line \(x = y + 1\) and between the lines \(y = -1\) and \(y = 1\).

Step 1: Graph the Boundary Line for the First Inequality

The first inequality is \(x - y < 1\), which is equivalent to \(x < y + 1\). First, graph the boundary line by rewriting it as an equation: \(x = y + 1\). This line has a slope of 1 and a y-intercept of -1. We draw this line as a dashed line on the graph since the inequality is <, indicating that points on the line are not included.

Step 2: Identify the Region for the First Inequality

Since the inequality is \(x < y + 1\), shade the region below the line \(x = y + 1\) because this is where the x-values are less than the values on the line. Remember that the line itself is not included in the solution set.

Step 3: Graph the Boundaries for the Second Inequality

For the second inequality \(-1 < y < 1\), graph the lines \(y = -1\) and \(y = 1\). Both lines should be drawn as dashed lines because the inequality does not include equality, i.e., it is strict.

Step 4: Identify the Region for the Second Inequality

Shade the region between the lines \(y = -1\) and \(y = 1\). This is the vertical strip where the y-values are between -1 and 1, not including the boundaries themselves.

Step 5: Determine the Solution Set

The solution set of the system of inequalities is the overlapping shaded region between the two individual solutions. This will be the area that is both below the line \(x = y + 1\) and between \(y = -1\) and \(y = 1\). Highlight this region as the final solution.

Key Concepts

Graphing Inequalities

Graphing inequalities involves plotting the boundary line associated with each inequality and identifying the region that satisfies the inequality condition. When graphing an inequality like \(x - y < 1\), you first convert the inequality to an equation, \(x = y + 1\), to find the boundary line. To determine the graph, you should plot the line and decide whether the region of interest lies above or below it.

Once the line is determined, remember:

• If the inequality uses \(<\) or \(>\), the line should be dashed to show that points on the line are not included in the solution set.
• If the inequality uses \(\leq\) or \(\geq\), the line should be solid, meaning points on the line are included in the solution set.

This distinction helps in accurately representing the solution on a graph.

Solution Set

The solution set involves finding the common area that satisfies all given inequalities simultaneously. Once you have plotted and shaded the regions for each inequality, their overlap is the solution set. In graphing systems of inequalities, this overlapping area is crucial, as it represents all possible solutions that satisfy every condition given.

To simplify:

• Identify the region shaded for each inequality.
• The solution set will be the intersection of these regions, where the colors overlap if using graphing tools or shading manually.

It is vital to highlight the solution set clearly, as it encapsulates the outcomes that obey all rules of the system.

Linear Inequalities

Linear inequalities are similar to linear equations but use inequality signs \(<, >, \leq, \geq\) instead of an equal sign. They define regions on the graph rather than just a single line. For example, \(-1 < y < 1\) describes a band between two horizontal lines, \(y = -1\) and \(y = 1\), both of which are not included due to the strict inequality.

This type of inequality divides the graph into sections:

• The region satisfying the inequality,
• The boundary lines depicted as dashed or solid, showing inclusion or exclusion, respectively,

Understanding and plotting these divisions accurately is key to solving and graphing systems of inequalities.

Boundary Lines

Boundary lines serve as the reference markers in graphing inequalities. They represent the points where the inequality exactly balances the equation, like \(x = y + 1\) for \(x - y < 1\).

Key points about boundary lines:

• Boundary lines are drawn as dotted if the relationship is strictly less than or greater than.
• If the inequality includes equality, the line will be solid to show that points on the line satisfy the inequality.

These boundaries help to fashion the "canvas" that models possible solutions, indicating where one must shade on either side to represent the possible area where the solutions to the inequality exist.