Problem 1 - Algebra for College Students

Question

For Problems \(1-24\), indicate the solution set for each system of inequalities by graphing the system and shading the appropriate region. $$ \left(\begin{array}{l} x+y>3 \\ x-y>1 \end{array}\right) $$

Step-by-Step Solution

Answer

The solution set is the region above \(y = -x+3\) and below \(y = x-1\), excluding the boundary lines.

Step 1: Write the Inequalities in Slope-Intercept Form

The first inequality is \(x + y > 3\). To write it in slope-intercept form, solve for \(y\): \(y > -x + 3\). The second inequality is \(x - y > 1\). Solve for \(y\) to get: \(y < x - 1\).

Step 2: Graph the Boundary Lines

For \(y = -x + 3\), the boundary line is a straight line with a slope of -1 and y-intercept of 3. For \(y = x - 1\), the boundary line has a slope of 1 and y-intercept of -1. Plot these lines on the coordinate plane. Use dashed lines because these are strict inequalities ( > or < ).

Step 3: Shade the Solution Regions

For \(y > -x + 3\), shade above the line \(y = -x + 3\). For \(y < x - 1\), shade below the line \(y = x - 1\). The overlapping region where both conditions are true is the solution set.

Key Concepts

Graphing Inequalities

Graphing inequalities involves representing regions on a coordinate plane that satisfy each inequality in a system. Inequalities are similar to equations but involve symbols like ">" and "<" instead of "=". Here's how we graph them:

• First, rewrite the inequalities in a form that is easy to graph, often the slope-intercept form.
• Second, identify the boundary line of each inequality. The boundary line is graphed based on the rewritten equation, using either a solid line (for inequalities with ≥ or ≤) or a dashed line (for strict inequalities with > or <).
• Third, shade the region of the graph that satisfies the inequality. This involves determining which side of the boundary line contains the solutions. For greater than (>) or greater than or equal to (≥) inequalities, shade above the line. For less than (<) or less than or equal to (≤) inequalities, shade below the line.

The final step is to identify where the shaded regions overlap. This overlapping area represents the solution set for the system of inequalities.

Slope-Intercept Form

The slope-intercept form is a way of writing linear equations, making them easy to understand and graph. It is given by the form:\[y = mx + b\] where:

• \(m\) is the slope, indicating how steep the line is and the direction in which it tilts. A positive slope means the line rises from left to right, while a negative slope means it falls from left to right.
• \(b\) is the y-intercept, representing the point where the line crosses the y-axis.

For example, in the inequality \(y > -x + 3\), the slope-intercept form tells us that the slope is -1 and the y-intercept is 3. Similarly, for \(y < x - 1\), the slope is 1 and the y-intercept is -1.Using this form helps in quickly sketching the line on a coordinate plane and determining which side to shade for inequalities, making it a crucial step in solving systems of inequalities.

Solution Set

The solution set of a system of inequalities is the region where all inequalities in the system overlap on the graph. This region represents all the coordinate points that satisfy each inequality simultaneously. Here's how you can determine the solution set:

• Start by graphing each inequality separately on the same coordinate plane. Use dashed or solid lines according to whether the inequality is strict or inclusive.
• Next, shade the region that satisfies each inequality.
• The solution set is the area where these shaded regions overlap. Every point within this region is a solution, meaning it satisfies all inequalities in the system.

Understanding the concept of a solution set is critical, as it visually and practically displays all the possible solutions a system of inequalities can have.

Coordinate Plane

A coordinate plane is a two-dimensional surface defined by a horizontal axis (x-axis) and a vertical axis (y-axis). This plane is divided into four quadrants and serves as the backdrop for graphing equations and inequalities. Here's what you need to know:

• Each point on the plane is defined by an ordered pair \((x, y)\), where \(x\) is the coordinate on the horizontal axis and \(y\) is the coordinate on the vertical axis.
• Graphing on a coordinate plane begins by plotting known points, such as y-intercepts, and using the slope to determine the direction and steepness of lines.
• For inequalities, the plane also aids in shading regions that represent possible solutions, helping to visually solve the system of inequalities.

The coordinate plane is a vital tool in mathematics that transforms abstract inequalities into a visual form, making it easier to comprehend and analyze their solutions.