Question

Linear Inequality

Answer 1

Answer:

Step-by-step explanation:

Temporarily substitute the = symbol in this given inequality:   3x - 4y = 8.

Now pick 2 or 3 y values and find the corresponding x value for each:

If y = 2, 3x - 4(2) = 8, so that 3x = 16 and x = 16/3.  Thus, one point on the graph is (16/3, 2).

If y = 0, 3x - 4(0) = 8, and so 3x = 8, or x = 8/3.  Another point on the graph is (8/3, 0).

Now plot these two points (8/3, 0) and (16/3, 2), and draw a solid line through them (because " ≥ " includes an " = " sign).

Finally, invent a 3rd point that is NOT on this solid line.  Substitute the x- and y- coordinates into the original inquality 3x - 4y ≥ 8.  Two possible outcomes are:  a) The inequality is true.  All solutions are on this side of the solid line you drew.  Shade that side of the line.  b) The inequality is false.  Do NOT shade that side of the line.

Note:  an easier way of graphing the solid line is described as follows:

Let x = 0 and find y:  3(0) - 4y = 8, so that y = -2 and a point on the line is (0, -2).

Then let y = 0 and find x:  3x = 8, or x = 8/3 and this corresponds to the point (8/3, 0) on the line.

Answer 2

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Answer:

  see below (shaded to the right of the solid line through (0, -2) and (4, 1))

Step-by-step explanation:

The boundary line of the solution area is the line ...

  3x -4y = 8

This can be plotted a number of ways. One is to find a couple of points on the line, then draw a line through them.

We notice that the coefficient 4 is a factor of the constant 8, so we can find one point by setting x=0:

  3·0 -4y = 8   ⇒   y = -2 . . . . . the y-intercept.

Another point can be found by setting x to some multiple of 4, such as 4.

  3·4 -4y = 8   ⇒   y = 1

So, two points on the boundary line are (0, -2) and (4, 1). Since the "or equal to" case is included in the inequality, the boundary line is drawn as a solid line.

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We notice that the variable x has a positive coefficient. Considering only that variable and the inequality symbol, we have x ≥ ( ). This means the solution area will be shaded for larger x values, those to the right of the boundary line.

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Additional comment

If we were to add 4y to the equation, it becomes 3x ≥ 4y +8. This makes the y-coefficient be positive. Considering the y-variable and the inequality symbol, we now have ( ) ≥ y. This indicates the solution set includes y-values below (less than) those on the boundary line. For this boundary line with positive slope, "below" is the same direction as "to the right" as far as the shading of the solution space is concerned.