Problem 27 - College Algebra

Question

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \begin{aligned}&y>2 x-3\\\&y<-x+6\end{aligned} $$

Step-by-Step Solution

Answer

The solution set of the given system of inequalities is the intersection of shaded regions from both graphs. This intersection represents all points that satisfy both inequalities.

Step 1: Graph the First Inequality

Start by graphing the inequality \( y > 2x - 3 \). Graph the line as if it was an equality \( y = 2x - 3 \). This will be a straight line with \(-3\) as y-intercept and a slope of \(2\). Because the inequality is 'greater than', you will use a dashed line to represent the graph of the line. After this, choose a test point – (0, 0) is the simplest one – and if it satisfies the inequality, shade that region. If it doesn't, shade the other region.

Step 2: Graph the Second Inequality

Next, graph the inequality \( y < -x + 6 \). Graph the line as if it was an equality \( y = -x + 6 \). This line also has y-intercept at \(6\) and a slope of \(-1\). Because the inequality is 'less than', again use a dashed line to represent the graph of the line. Choose a test point (like 0,0) and if it satisfies the inequality, shade that region. If it doesn't, shade the other region.

Step 3: Find the Solution Set

Now you have two graphs on the same coordinate plane. The solution set for the system of inequalities is the intersection of these two shaded regions. Therefore, look for the region that is shaded twice. This area is the solution set for the system of inequalities.

Key Concepts

Graphing Inequalities

Graphing inequalities involves drawing a line on the coordinate plane to represent the boundary of the inequality. First, you treat the inequality as an equation to find the line. For example, the inequality \(y > 2x - 3\) becomes the line \(y = 2x - 3\). The line is drawn using the standard y-intercept and slope method. The y-intercept is where the line crosses the y-axis, and the slope tells you the angle of the line.

• The y-intercept for \(y = 2x - 3\) is \(-3\).
• The slope is \(2\), which means for every increase of 1 in \(x\), \(y\) increases by 2.

Since \(y > 2x - 3\) doesn't include equal points (it's 'greater than', not 'greater than or equal to'), the line is dashed to show the boundary isn’t included. After drawing the line, choose a test point like \( (0, 0) \) to decide which side to shade. Shading shows the area of the plane that satisfies the inequality. This location-changing shading indicates multiple points where the inequality holds true.

Solution Set

The solution set of a system of inequalities is the intersection of shaded areas that satisfy each inequality. Once the graph for each inequality is drawn and shaded, find where both shaded areas overlap. This overlapping region is where both inequalities are true at the same time.

• The solution set is where both conditions \(y > 2x - 3\) and \(y < -x + 6\) hold.
• This area is typically highlighted or cross-hatched.

To consider a point within this region the true solution of the system, it must satisfy all inequalities in the system. Thus, the solution set forms a kind of "zone of truth" within the coordinate plane, where all parts of the system mesh together harmoniously.

Coordinate Plane

The coordinate plane is a two-dimensional surface where each location is defined by a pair of numbers, \((x, y)\). This grid-like format is crucial for graphing equations and inequalities in a visually understandable way. The horizontal axis is the x-axis, and the vertical axis is the y-axis.

• The intersection point of these axes is called the origin, at \((0, 0)\).
• When graphing, you plot points or lines on this grid to visually represent mathematical concepts.

Each line you draw in relation to inequalities helps split the coordinate plane into regions that either satisfy or don’t satisfy the inequality. Visualizing these regions allows for quick identification of possible solutions to systems of inequalities.

Test Points

Test points are specific spots on the coordinate plane you pick to determine which region of the plane satisfies an inequality. Usually, simple points like \((0, 0)\) are chosen, as they make calculations easy and reduce errors when evaluating whether they satisfy the inequality.

• For \(y > 2x - 3\), plug \((0, 0)\) into the inequality: \(0 > 2(0) - 3\), simplifies to \(0 > -3\), which is true.
• Therefore, the region containing \((0, 0)\) is shaded for this inequality.

If a test point satisfies the inequality, shade the region containing that point; if not, shade the opposite region. By using test points, you easily determine which parts of the graph represent solutions without having to manually check every possible point, making solving inequalities more efficient.