Algebra II HW Questions (Please show step by step and explain)

Question

4. Graph the area bounded by

 y< 1/2x+6, X+3y>=12,  X>=0 and x<=12

 

part i. graph the bounded by X>=0 and x<=12. shade the region that satisfies both inequalites 

 

part ii. to graph you've already produced, add the region bounded by the lines y<1/2x+6 and x+3y>=12. shade the region that satisfies both of these inequalties and the ones in part I. 

 

 

 

(Thank u!)

Answer 1

On a coordinate grid, draw vertical lines on the y-axis and at x=12.  One the same coordinate grid, graph the line y=(1/2)x+6 using dashed lines and shade the region below the dashed lines.  Graph the second line y=(-1/3)x+4 using a solid line and shade the region above that line.

 

Be sure to shade the region between the y-axis and the vertical line at x=12.  The solution is the shaded region that satisfies all the conditions.

Answer 2

GRAPHING LINES AND LINEAR INEQUALITIES

Given a linear equation, you can “solve for y”. This may involve transposing terms. Even when you have an inequality, you can still “solve for y”.

You have your equation or inequality in the form or or the like. Now you know the y-intercept (it’s “b”) and your line (or boundary line) passes through that point on the y-axis.

To draw the line, you need to find the x-intercept. Use and set y = 0. Now solve for x; this value is the x-intercept. Find that point on the x-axis, and draw your line connecting these two axis intercepts.

For an inequality’s boundary line, draw it as a dashed line if the inequality does not include the = case.

When you deal with an inequality, you shade the half-plane ABOVE the boundary line, if the inequality was of the y > (or ) type; otherwise BELOW for the y < (or ) type.


Note: Using this method means that there is no need for a ‘test point’ as taught in the textbook.

When an inequality is of the form x > 5, the boundary line is vertical, and the shaded half-plane is to the right of the dashed vertical line passing through 5 on the x-axis. For x < -2, the shading is to the left of the dashed vertical line passing through -2 on the x-axis.

 

Each individual linear inequality gives a solution (a half-plane). When you have a system of linear inequalities, the solution to the system is the INTERSECTION of all half planes, i.e. where the various half planes overlap.