Problem 31 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The system of inequalities \(y > 2x - 3\) and \(y < -x + 6\) intersect at the region above the line \(y = 2x - 3\) and below the line \(y = -x + 6\). This is the solution to the system of inequalities.

Step 1: Graph the first inequality

The first inequality is \(y > 2x - 3\). This can be graphed by first sketching the line \(y = 2x - 3\) on the coordinate plane. This line has a slope of 2 and a y-intercept of -3, so it cuts the y-axis at -3 and has a gradient of 2. Because the inequality sign is a 'greater than' symbol, the solution set consists of all points above the line \(y = 2x - 3).\)

Step 2: Graph the second inequality

The second inequality is \(y < -x + 6\). This line can be sketched on the same set of axes by drawing the line \(y = -x + 6\). This line has a slope of -1 and a y-intercept of 6, so it cuts the y-axis at y=6. Because the inequality is 'less than', the solution to this inequality includes all points below the line \(y = -x + 6\).

Step 3: Determine the solution set

The solution to the system of inequalities is the region where the solution sets to both inequalities overlap. This can be identified by finding the region that is both above the line \(y = 2x - 3\) and below the line \(y = -x + 6\).

Key Concepts

Graphing Inequalities

Graphing inequalities involves drawing lines on a coordinate plane to represent the set of solutions for each inequality. For the first inequality, we start by plotting the equation in its equality form, such as transforming the inequality from \( y > 2x - 3 \) to \( y = 2x - 3 \). The line created is the boundary of the inequality. If the inequality symbol is 'greater than' or 'less than', you use a dashed line to indicate that points on the line itself are not included. If it were 'greater than or equal to' or 'less than or equal to', you would use a solid line.Additional details to consider are:

• Slope: The first line has a slope of 2, creating a steep incline, while the second line’s slope is -1, creating a downward slope.
• Y-intercept: For the first equation, the line crosses the y-axis at \( y = -3 \) and six for the second equation.

These elements guide how the lines are placed on the plane.

Solution Set

The solution set is the region of the coordinate plane where both inequalities in the system are true simultaneously. After graphing each line, the solution set is typically the area that lies above or below these boundaries, depending on the inequalities' direction.To solve the given inequality system:

• Shade above the line \( y = 2x - 3 \) for the inequality \( y > 2x - 3 \).
• Shade below the line \( y = -x + 6 \) for the inequality \( y < -x + 6 \).

The overlapping region of these two shaded areas represents the solution set where both conditions hold true. There should be a clear, distinct region on the graph where all points satisfy both equations. This is where the magic of systems of inequalities comes to life, by visualizing how solutions to multiple conditions intersect.

Coordinate Plane

The coordinate plane is a two-dimensional surface where you can graph equations and inequalities. It consists of two axes intersecting at a point called the origin.

• X-axis: The horizontal axis.
• Y-axis: The vertical axis.

Using these axes, you can plot points, lines, and regions.For this exercise:

• First, identify the intersections where the lines cross the y-axis \((0, -3)\) and \((0, 6)\).
• Then plot another point using the slopes. For \( 2x - 3 \), another point could be \((1, -1)\), so the line passes through these two points.
• For \(-x + 6 \), another point could be \((1, 5)\).

Draw these boundary lines accordingly, ensuring precision to get an accurate representation of each inequality.

Inequality Graphing

Inequality graphing blends the techniques of algebra and geometry, transforming algebraic inequalities into visual region definitions. It requires understanding both the procedural steps and visual representation.When graphing:

• Ensure to determine the type of inequality to choose the correct line style - dashed for strict inequalities and solid for inclusive ones.
• Consider using test points to verify which side of the line satisfies the inequality. For example, substitute a point like \((0,0)\) to see if it adheres to the inequality.
• Draw attention to where the two regions overlap—only this area is part of the solution set.

This graphical method is immensely powerful, offering a direct view into multiple solution regions. Students often find the visual representation helpful in understanding how solutions can intertwine and where exact satisfaction occurs.