Question

solve the following system of inequalities graphically on the set of axes below. the coordinates of a point in the solution set. $y > -x + 4$ $y < 3x - 4$

Explanation:

Step1: Find intersection of two lines

To find the intersection of \( y=-x + 4 \) and \( y = 3x-4 \), set them equal:
\( -x + 4=3x - 4 \)
\( 4 + 4=3x+x \)
\( 8 = 4x \)
\( x = 2 \)
Substitute \( x = 2 \) into \( y=-x + 4 \), we get \( y=-2 + 4=2 \). So the intersection point is \( (2,2) \).

Step2: Determine the solution region

For \( y>-x + 4 \), the region is above the line \( y=-x + 4 \) (dashed line). For \( y<3x - 4 \), the region is below the line \( y = 3x-4 \) (dashed line). The solution set is the overlapping region of these two.

Step3: Find a point in the solution set

We can test points. Let's take \( x = 3 \). For \( y>-x + 4 \), when \( x = 3 \), \( y>-3 + 4=1 \). For \( y<3x - 4 \), when \( x = 3 \), \( y<9 - 4 = 5 \). So we can choose \( y = 2 \) (satisfies both \( 2>1 \) and \( 2<5 \)). So the point \( (3,2) \) is in the solution set. (We can also see from the graph, the overlapping region, and pick a point there. Another example: \( (4,3) \), check \( 3>-4 + 4=0 \) (yes) and \( 3<12 - 4 = 8 \) (yes).)

Answer:

A point in the solution set is \( (3,2) \) (or other valid points like \( (4,3) \), \( (2.5,3) \) etc.)