Find the coordinates of any vertices formed. 3x + 4y ≥ 12, 5x + 6y ≤ 30, and 1 ≤ x ≤ 3.
I tried graphing this, which formed a parallelogram, but it was hard to find eyeball vertices.
That's what algebra is for. Pair up the equations to fin the intersections:
3x+4y = 12
5x+6y = 30
x = -16 and y = 15
and likewise for the others. No eyeballing needed.
Shouldn't it have 4 points due to 4 vertices?
To find the coordinates of the vertices formed by the given inequalities, you can solve the system of equations formed by their intersections.
Let's start by finding the intersection points between the first two inequalities:
1. Start with the inequality 3x + 4y ≥ 12:
- Rewrite it in slope-intercept form by isolating y: y ≥ -(3/4)x + 3.
- Draw the line y = -(3/4)x + 3.
2. Next, consider the inequality 5x + 6y ≤ 30:
- Rewrite it in slope-intercept form by isolating y: y ≤ -(5/6)x + 5.
- Draw the line y = -(5/6)x + 5.
3. Now, find the intersection point of the two lines:
- Set the two equations equal to each other and solve for x: -(3/4)x + 3 = -(5/6)x + 5.
- Simplify the equation and solve for x: (3/4)x - (5/6)x = 5 - 3.
- Continue simplifying and solving: (9/12)x - (10/12)x = 2/12, -x/12 = 2/12, x = -2.
- Substitute x = -2 into either equation to find y: y = -(3/4)(-2) + 3, y = (3/2) + 3, y = 4.5.
- The intersection point is (-2, 4.5).
4. Now, check if the intersection point lies within the given range for x (1 ≤ x ≤ 3). Since x = -2 is outside the range, discard this point.
Repeat the above steps to find the intersection points between the other combinations of inequalities:
For 3x + 4y ≥ 12 and 1 ≤ x ≤ 3:
- Check if any points intersect within the given range. If they do not, discard the point.
For 3x + 4y ≥ 12 and 5x + 6y ≤ 30:
- Check if any points intersect. If they do, keep those points.
For 5x + 6y ≤ 30 and 1 ≤ x ≤ 3:
- Check if any points intersect within the given range. If they do, keep those points.
Once you have identified the intersection points that satisfy the given range, those points will be the vertices of the region formed by the inequalities.
Note: Solving inequalities algebraically and graphing are both valid methods, and sometimes graphing is easier to visualize the solution. However, finding the exact coordinates of the vertices may require more precise algebraic calculations.