Hints for solving the problems (4.3)

Problems 1-4.

In all these problems substitute the order pair (x y) into each inequality and if it makes a true statement put a check. If it doesn’t leave it blank.

Problem No. 1   -    (2, -1) (4, -2) (-5, 2) (3, 8) (-3, 1) (4, 3) (-1, 6)

     x - y <1      2+1<1      4+2<2      -5-2<2      3-8<2      -3-1<2      4-3<1      -1-6<2

                          no            no            yes         yes          yes          no            yes

Watch out for an inequality such as:   -2+1≤-1. This is true as the inequality is read as: less than or equal to, and -1 is less than or equal to -1.

Problems 5-28

Problem No. 5 – Graph the inequality    x - y > 4

In all these problems, graph the equality first and draw the line. If you have < or > the line will be open (broken). If you have ≤ or ≥ the line will be solid. The answer will shaded as regions or half planes.

Graph the equality:      x - y = 4      -y = -x + 4     y = x - 4

                               m = 1/1      b = -4

Plot the point (0, -4). From that point go one unit to the right and one unit up. You will land on the point (1, -3). Draw an open (broken) line between the two points because of the > in the problem.

Then substitute a point into the original inequality. Use the point (0, 0). This point can be used except when the line passes through it. Then use another point such as: (1,0).

       x - y > 4      Use the point (0, 0)      0 - 0 > 4      0 > 4      true or false

False, so you shade the region that doesn’t contain the point (0, 0). This region on the graph is down and to the right of the broken line.

Problem No. 21 – Graph the inequality      -4x - y ≤ -3.      Graph the equality

                -4x - y = -3        -y = 4x + -3      y = -4x + 3

                       So the   m = -4/1   and   b = 3

Plot the point (0, 3) and move one unit to the right and 4 units down and land on the point (1, -1). Draw a solid line between the two points, because of the ≤ in the inequality.

    Now substitute (0, 0) into the original inequality.

           -4x - y ≤ -3      -4(0) - 0 ≤ -3      0 - 0 ≤ -3      T or F

0 ≤ -3 False, shade the region that doesn’t contain the point (0, 0). That region is to the right and up.

Problem No. 26 – Graph the inequality   -3x + (1/3)y ≥ -2     Graph the equality                   -3x + (1/3)y = -2      Multiply by 3 to get rid of the fraction.

        3(-3x) + 3(1/3)y = 3(-2)       -9x + y = -6       y = 9x - 6

                             m = 9/1      b = -6

Plot the point (0, -6) and move one unit to the right and 9 units up and you land on the point (1, 3). Draw a solid line between the two points, because of the ≥ in the inequality.

    Now substitute (0, 0) into the original inequality.

        -3x + (1/3)y ≥ -2      -3(0) + (1/3)(0) ≥ -2      0 + 0 ≥ -2      0 ≥ -2

True or False  True, so you shade the region of the graph that contains the point (0, 0). This region is to the left and up from the line.