Problem 62 - Algebra 2

Question

Solve each system of inequalities. \(x+y \leq 9\) \(x-y \leq 3\) \(y-x \geq 4\)

Step-by-Step Solution

Answer

The solution is the region of overlap from graphing the inequalities.

Step 1: Graph the Boundary Lines

Firstly, transform each inequality into an equation to graph the boundary lines. For \(x+y = 9\), \(y = 9-x\). For \(x-y = 3\), \(y = x-3\). For \(y-x = 4\), \(y = x+4\). Plot these lines on the coordinate plane.

Step 2: Determine Shaded Regions

For each inequality, decide which side of the boundary line to shade. For \(x+y \leq 9\), shade below the line \(y = 9-x\). For \(x-y \leq 3\), shade below the line \(y = x-3\). For \(y-x \geq 4\), shade above the line \(y = x+4\).

Step 3: Find the Intersection of Shaded Regions

Find where the shaded regions of the inequalities overlap on the graph. This region represents the solution to the system of inequalities.

Key Concepts

Graphing Inequalities

Graphing inequalities involves converting each inequality into an equation, graphing the resulting lines, and then shading the appropriate regions on the graph. Graphing each inequality in a system of inequalities helps us visualize the solutions. To start, rewrite each inequality in slope-intercept form, where it looks like this:

• For example, take the inequality \(x + y \leq 9\), rewrite it as \(y \leq 9 - x\).
• Similarly, \(x - y \leq 3\) becomes \(y \leq x - 3\), and \(y - x \geq 4\) translates to \(y \geq x + 4\).

Once converted, you can plot each as an equation by drawing boundary lines on the coordinate plane. The line represents all the points where the inequality equals zero. Remember that the way you draw your line (solid or dashed) depends on the inequality sign:

• Use a solid line if the inequality includes \(\leq\) or \(\geq\), as points on the line satisfy the inequality.
• Use a dashed line for \(<\) or \(>\), as points on this line do not satisfy the inequality.

Coordinate Plane

The coordinate plane, often called the Cartesian plane, is a two-dimensional surface defined by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). Where these axes intersect is called the origin.

• The x-axis allows for plotting horizontal positions.
• The y-axis helps to plot vertical positions.

Each point on the plane is described by a pair of numbers, known as coordinates. These coordinates are written as an ordered pair, (x, y).

• The first number indicates the position along the x-axis.
• The second number tells the position along the y-axis.

When plotting points or lines, knowing how to navigate the coordinate plane is essential. You will use these axes to mark points precisely and graph lines. In the context of linear inequalities, the lines plotted based on equations will help visualize the solution sets of these inequalities on the coordinate plane.

Shaded Region

In the context of graphing inequalities, shaded regions on a coordinate plane show all possible solutions that satisfy a given inequality. Once the boundary lines are drawn, it's time to decide which part of the plane should be shaded for each inequality:

• For \(x + y \leq 9\), shade below the line \(y = 9 - x\).
• For \(x - y \leq 3\), shade below the line \(y = x - 3\).
• For \(y - x \geq 4\), shade above the line \(y = x + 4\).

The shading direction tells us where the solutions to each inequality lie. To decide on the side to shade, you can pick a test point, commonly the origin (0,0), and see if it satisfies the inequality:

• If (0,0) makes the inequality true, shade the region containing this point.
• If it does not, shade the opposite side.

The graph's solution to a system of inequalities is the intersection of all shaded regions, which means it includes all the points that fulfill all conditions simultaneously. This region visually represents all possible solutions.