Guided Example #3 - Inequalities

Guided Example

(SMR 1.2a Graphs of Linear Inequalities) The boundaries of two inequalities intersected at (4, -3). The slope of one boundary is 4. The slope of the other boundary is the negative reciprocal of the first boundary. Both boundaries are part of the solution. (0, -5) is a point in the overlapping region defined by the inequalities. Which of the following systems best describes these inequalities?

A. ${\begin{cases} 4 x - y \geq 19 \\ x + 4 y \geq - 8 \end{cases}$ B. ${\begin{cases} 4 x - y \leq 19 \\ x + 4 y \leq - 8 \end{cases}$ C. ${\begin{cases} 4 x - y \geq 2 \\ x + 4 y \geq 76 \end{cases}$ D. ${\begin{cases} 4 x - y \leq 19 \\ 4 x + y \leq - 2 \end{cases}$

The correct answer is B.

Step 1

Substituting the point of intersection into the slope intercept form, $y = 4 x + b - \frac{1}{4} \Rightarrow (- 3) = 4 (4) + b \Rightarrow$ $b = 19$

Step 2

The equation of the first boundary is $y = 4 x - 19$ The slope of the second boundary is $- \frac{1}{4}$

Step 3

Substituting the point of intersection into the slope intercept form, $y = - \frac{1}{4} x + b \Rightarrow (- 3) = - \frac{1}{4} (4) + b \Rightarrow b = - 2$ . The equation for the secondary boundary is $y = - \frac{1}{4} x - 2$ .

Step 4

Now we must find the region that satisfies this system by testing the given point (0, -5)
First boundary:

$y = 4 x - 19 \Rightarrow - 5 = 4 (0) - 19 \Rightarrow - 5 \geq - 19$ thus

$y \geq 4 x - 19$
Second boundary:

$y = - \frac{1}{4} x - 2 \Rightarrow - 5 = - \frac{1}{4} (0) - 2 \Rightarrow - 5 \leq - 2$ .

Step 5

Thus $y \leq - \frac{1}{4} x - 2 {\begin{cases} y \geq 4 x - 19 \\ y \leq - \frac{1}{4} x - 2 \end{cases}$ changed to standard form is ${\begin{cases} 4 x - y \leq 19 \\ x + 4 y \leq - 8 \end{cases}$

All Steps