Problem 23 - Intermediate Algebra

Question

For Problems \(17-32\), indicate the solution set for each system of inequalities by shading the appropriate region. $$ \left(\begin{array}{l} y

Step-by-Step Solution

Answer

The solution set is the region between the lines \(y = x\) and \(y = x + 1\), above \(y = x\) and below \(y = x + 1\).

Step 1: Understand the Inequalities

The system of inequalities given is: \( y < x + 1 \) and \( y \geq x \). We need to graph these inequalities to find the solution region.

Step 2: Graph the First Inequality

Graph the first inequality \( y < x + 1 \). First, graph the line \( y = x + 1 \), which is a straight line with a slope of 1 and y-intercept at 1. Use a dashed line because the inequality is strict meaning it does not include the line. Shade the area below this line.

Step 3: Graph the Second Inequality

Graph the second inequality \( y \geq x \). First, graph the line \( y = x \), which is a straight line with a slope of 1 and y-intercept at 0. Use a solid line as the inequality includes the line. Shade the area above this line.

Step 4: Determine the Solution Region

The solution region is where the shaded area from both inequalities overlaps. This will be the region above the line \( y = x \) and below the line \( y = x + 1 \).

Step 5: Verify the Solution Region

Choose a test point from the overlapping region to verify, such as \((0.5, 0.5)\). Substitute it into both inequalities:For \( y < x + 1 \): \(0.5 < 0.5 + 1\) is true.For \( y \geq x \): \(0.5 \geq 0.5\) is true.Both are satisfied, confirming the validity of the solution set.

Key Concepts

Inequality Graphing

Graphing inequalities is an essential skill in algebra that helps visualize solutions. To graph an inequality, you first plot the associated equation as if it were an equality. For instance, in the inequality \( y < x + 1 \), you begin by sketching the line \( y = x + 1 \). This line is your boundary line, which visually represents where the possible solutions could be. Since the inequality is strict (\(<\) or \(>\)), use a dashed line to indicate that points on the line are not part of the solution.After plotting the line, decide which side of the line to shade. If the inequality is \(<\) or \(>\), shade below or above the line respectively. This shaded area represents all the possible solutions that make the inequality true.

Solution Region

The solution region in a system of inequalities is the area where the shaded regions of the inequalities overlap. When graphing two or more inequalities, each inequality contributes a shaded area to the graph. The solution region is the intersection of these areas. For example, the system \( y < x + 1 \) and \( y \geq x \) results in two lines being drawn.- \( y = x + 1 \) (dashed line, shading below)- \( y = x \) (solid line, shading above)The overlapping region forms a band between these two lines. This band indicates the set of all points that satisfy both inequalities simultaneously. It's where you will find solutions that meet the criteria of every inequality given in the system.

Linear Inequalities

Linear inequalities involve expressions where variables are only raised to the first power, and you work with inequality symbols like \(<\), \(>\), \(\leq\), and \(\geq\). When you encounter a system of linear inequalities, the process includes dealing with equations of the form \( ax + by < c \), \( ax + by > c \), or their non-strict counterparts.Linear inequalities can be conceptualized as an extension of linear equations but with a key difference: instead of a single line, you're considering a half-plane composed of all points on one side of that line. This often translates graphically to shading a region on a coordinate plane, representing an infinite set of solutions.Understanding linear inequalities' structure allows you to solve systems by identifying where these half-planes intersect. This intersection is crucial because it showcases the solution set applicable to all inequalities considered together.