Problem 115 - College Algebra Essentials

Question

Sketch the graph of the solution set for the following system of inequalities: $$\left\\{\begin{array}{l}y \geq n x+b(n<0, b>0) \\\y \leq m x+b(m>0, b>0)\end{array}\right.$$

Step-by-Step Solution

Answer

The graph of the solution set forms a wedge shape with the two lines \(y = mx + b\) and \(y = nx + b\) as its boundaries. The solution set of the system of inequalities are the common shaded area between the two inequalities.

Step 1: Interpret the inequalities

There are two inequalities given. \(y \geq nx + b\) and \(y \leq mx + b\) . Both represent a straight line with slopes \(n\) and \(m\) respectively. The inequality sign indicates whether the area of interest is above or below the drawn line.

Step 2: Plot the line \(y = nx + b\)

Plot the line \(y = nx + b\) which has a negative slope 'n'. This will be a straight line slanting diagonally down to the right starting from the y-intercept 'b'.

Step 3: Identify the area for the first inequality

Since the inequality is \(y \geq nx + b\), the solution set lies above the line, so shade the region above the line.

Step 4: Plot the line \(y = mx + b\)

Next, plot the line \(y = mx + b\) which has a positive slope 'm'. This will be a straight line climbing diagonally to the right from the y-intercept 'b'.

Step 5: Identify the area for the second inequality

Since the inequality is \(y \leq mx + b\), the solution set lies below the line, so shade the area below the line.

Step 6: Identify common shaded area

The common shaded area between the two inequalities represents the solution set to the system of inequalities. This region is where all solutions that satisfy both inequalities at the same time are found.