Question

3. Solve the following system of inequalities graph

on the set of axes below.
3x + y <7
X- - 4
State the coordinates of a point in the solution set.

Answer 1

To solve the system of inequalities, graph the equations on the set of axes and find the region where both inequalities are true. The solution set can be represented by shading the region below the line corresponding to the inequality. The coordinates of a point in the solution set can be found by choosing a point below the line.

Step-by-step explanation:

To solve the system of inequalities, you need to graph the equations on the set of axes and find the region where both inequalities are true.

First, graph the line 3x + y = 7. To do this, find two points that lie on the line by choosing values of x and solving for y. For example, let x = 0, then y = 7. Let x = 2, then y = 1. Plot these points and draw a line through them.

Next, shade the region below the line to represent the inequality 3x + y < 7. This region is the solution set. To find a point in the solution set, choose a point below the line, such as (1, 1).

Answer 2

see attached

Step-by-step explanation:

The boundary line of the first inequality has a slope of -3 and a y-intercept of 7. It is a dashed line because it is not included in the solution set. y-values in the set are less than those on the line, so shading is below it.

The boundary line of the second inequality has a slope of 2/3 and a y-intercept of -4. It is a solid line because it is included in the solution set. y-value in the set are greater than or equal to those on the line, so shading is above it.

The solution set is in the left quadrant of the X that the lines make. Any point in that quadrant will be in in the solution set. We have shown (-3, -2) on the graph. (0, 0) is another point in the solution set.

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Additional comment

When the inequality symbol is replaced with an equal sign (=), the resulting equation is the equation of the boundary line of the solution set. When that equation is written in the form ...

y = mx +b

the coefficient of x (m) is the slope of the line. It tells the rise/run of the line, the number of grid squares up for each square to the right. The constant (b) is the point on the y-axis where the line crosses. These values are usually sufficient to help you plot the line.

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To determine shading, pick a variable term with a positive coefficient. (In this set of inequalities, they all have positive coefficients.) Look at the relation of the term to the inequality symbol. This tells you where the shading is.

3x < . . . . shading is left of the line

y < . . . . . shading is below the line

The first line has negative slope, so "left" and "below" are on the same side of the line.

y ≥ . . . . shading is above the line

≥ 2/3x . . . . shading is left of the line (x is less than or equal values on the line)

The second line has positive slope, so "above" and "left" are the same side of the line.