Question

linear inequality systems graphically
question
video examples
solve the following system of
inequalities graphically on the set of
axes below. state the coordinates of a
point in the solution set.
( y > -x + 4 )
( y < 3x - 4 )
(here is a coordinate plane with x from -10 to 10 and y from -10 to 10)
answer
attempt 1 out of 2
point: ( , ) submit answer

Explanation:

Step1: Find Intersection of Lines

First, find the intersection of \( y = -x + 4 \) and \( y = 3x - 4 \). Set \( -x + 4 = 3x - 4 \). Solving: \( 4 + 4 = 3x + x \) → \( 8 = 4x \) → \( x = 2 \). Substitute \( x = 2 \) into \( y = -x + 4 \): \( y = -2 + 4 = 2 \). So intersection is \( (2, 2) \).

Step2: Determine Solution Region

For \( y > -x + 4 \), the region is above the line \( y = -x + 4 \) (dashed line, since \( > \)). For \( y < 3x - 4 \), the region is below the line \( y = 3x - 4 \) (dashed line, since \( < \)). The solution is where both regions overlap, to the right of the intersection point (since lines have slopes -1 and 3, the overlapping region is where \( x > 2 \), \( y \) between the lines).

Step3: Choose a Point in Solution

Pick a point in the overlapping region. Let's take \( x = 3 \). For \( y > -3 + 4 = 1 \) and \( y < 3(3) - 4 = 5 \). So \( y \) can be 2, 3, 4. Let's choose \( (3, 3) \). Check: \( 3 > -3 + 4 = 1 \) (true), \( 3 < 9 - 4 = 5 \) (true). So \( (3, 3) \) is in the solution set.

Answer:

Point: \( (3, 3) \) (other valid points: \( (4, 4) \), \( (3, 4) \), etc. are also correct)