Math

Question Graph the solution set of the linear inequalities system: x+4y8x + 4y \leq 8 and yx2y \geq x - 2.

Studdy Solution

STEP 1

Assumptions
1. We have a system of two inequalities: a. x+4y8x + 4y \leq 8 b. yx2y \geq x - 2
2. We need to graph the solution set of the system.
3. The solution set will be the region on the graph where both inequalities are satisfied simultaneously.
4. The boundary lines for the inequalities will be included if the inequality is less than or equal to (\leq) or greater than or equal to (\geq), and not included if it is strictly less than (<<) or strictly greater than (>>).

STEP 2

First, we will graph the boundary line for the first inequality x+4y=8x + 4y = 8. To do this, we will find the intercepts of the line.
For the x-intercept, set y=0y = 0: x+4(0)=8x + 4(0) = 8 x=8x = 8
For the y-intercept, set x=0x = 0: 0+4y=80 + 4y = 8 y=2y = 2

STEP 3

Plot the x-intercept (8, 0) and the y-intercept (0, 2) on the graph, and draw the line that passes through these points. This line represents the equation x+4y=8x + 4y = 8.

STEP 4

Since the inequality is x+4y8x + 4y \leq 8, we will shade the region below the line because this region represents all the points (x,y)(x, y) that satisfy the inequality.

STEP 5

Next, we will graph the boundary line for the second inequality y=x2y = x - 2. This is a linear equation with a slope of 1 and a y-intercept of -2.

STEP 6

Plot the y-intercept (0, -2) on the graph. To find another point, you can use the slope. Since the slope is 1, for every increase of 1 in x, y also increases by 1.

STEP 7

From the y-intercept (0, -2), move one unit to the right (which increases x by 1) and one unit up (which increases y by 1). This gives us the point (1, -1). Plot this point on the graph.

STEP 8

Draw the line that passes through the points (0, -2) and (1, -1). This line represents the equation y=x2y = x - 2.

STEP 9

Since the inequality is yx2y \geq x - 2, we will shade the region above the line because this region represents all the points (x,y)(x, y) that satisfy the inequality.

STEP 10

The solution set of the system of inequalities is the region where the shadings from both inequalities overlap. Identify this region on the graph.

STEP 11

Make sure to use a dashed line for the boundary lines if the inequality does not include equality (is strictly less than or greater than), and a solid line if it does include equality (is less than or equal to or greater than or equal to). In this case, both boundary lines should be solid because both inequalities include equality.

STEP 12

Label the axes and provide a clear indication of the shaded region that represents the solution set to the system of inequalities.
The solution set is the region that is shaded by both inequalities on the graph.

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