Answer to Question #103824 in Algebra for tyler

Question #103824
Graph the following and then answer the questions given below: 2x-3y Greater than or equal to 6

a) Describe the method you used to graph this inequality.

b) Where does the line cross the x-axis(write as an ordered pair)?

c) Where does the line cross the y-axis(write as an ordered pair)?

d) Is the origin on the right side or the left side of the line?

e) Is the line solid or dashed?

f) Which side of the line is shaded (left side or right side)?

g) How did you determine which side of the line to shade?
1
Expert's answer
2020-03-19T17:43:17-0400

a) The procedure to draw the straight line 2x3y62x-3y\ge6 .

We draw the straight line

2x3y=62x-3y =6 ....(1)

by reducing it into y=mx+cy=mx+c

Equation (1) can be written as 3y=2x63y=2x-6

Dividing with '3' we get

y=23x2y= \frac{2}{3}x-2

\Rightarrow slope m=23m= \frac{2}{3}

and y intercept c=2c= -2

Since slope m=riserunm= \frac{rise}{run} , for a rise by 2 units, run will be 3

\Rightarrow (3,0)(3,0) is a point on the straight line.

Since y intercept c=2c=-2 , (0,2)(0,-2) is a point on the straight line.

Joining these two points and extending it we get a straight line 2x3y=62x-3y=6 .

Since 2x3y62x-3y\ge6 is satisfied by the points x3x\ge3 and y2y\le-2

The equation is the points satisfying x3x\ge3 and y2.y\le-2.

c) Substituting x=0x=0 in the straight line we get 2(0)3y=62(0)-3y=6

y=2\Rightarrow y=-2

(0,2)(0,-2) is a point of intersection with y axis.

b) The point of intersection with xx axis is given by substituting y=0y=0 in the straight line.

2x3(0)=6\Rightarrow 2x-3(0)=6

\Rightarrow x=3x=3

\Rightarrow (3,0)(3,0) is the point of intersection with xx axis.

Therefore, the points of intersection with axes are (3,0)(3,0) and (0,2).(0,-2).

d) Since the straight line cuts the x axis at x=3x=3 and y axis at y=2y=-2 and 2x3y62x-3y\ge6 , the origin (0,0)(0,0) lies to the left of the straight line.

e) Since the straight line includes "equality", the points satisfying 2x3y=62x-3y=6 are also to be graphed.

Hence it is a solid line.

f) Right side of the line to be shaded.

g) Since 2x3y62x-3y\ge6 satisfied by x3x\ge3 and y2y\le-2 .





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