Hints for solving the problems - Apply 2.4

Steps for solving inequalities:

1. Add or subtract the same number from both sides of the inequality

will not change the direction of the inequality.

2. Multiply or divide the same positive number by both sides of the

inequality will not change the direction of the inequality.

3. Multiply or divide the same negative number by both sides of the

inequality will change the direction of the inequality.

Problem No. 2   –   Solve for y:   y  +  7  ≥  9

                                                            - 7      -7     Subtract 7 from both sides.

                                                          y    ≥    2

For the answer to any inequality problem, you need to put the solution set and the graph.

Solution set: {y│y ≥ 2} This is called the set builder notation. It reads y such that (│) y is greater than or equal to 2.

Graph: The graph is on the number line.

There is a solid dot on 2 and a shaded arrow to the right.

Problem No. 6  –  Solve for a:    4a   ≤   36

                                                       4          4      Divide both sides by 4.

                                                       a    ≤    9      Dividing by a positive number

                                                                                doesn’t change the direction.

      Solution set: {a│a ≤ 9}

Graph: The graph is on the number line.

There is a solid dot on 9 and a shaded arrow to the left.

Problem No. 10  –  Solve for k:    -3k   <   -9

                                                         -3        -3      Divide by -3.

                                                           k    >   3     By dividing by a negative

                                                                              number, the direction of the

                                                                               inequality is turned around.

    Solution set: {k│ k > 3}

Graph: The graph is on the number line.

There is an open circle on 3 and a shaded arrow to the right.

Problem No. 14.  –  Solve for m:    6m  -  8  >  -32

                                                                    8          8      Add 8 to both sides.

                                                          6m         >   -24

                                                            6                  6       Divide by 6.

                                                             m      >       -4

     Solution set: {m│m > -4}

Graph: On the number line, an open circle

at -4 and a shaded arrow to the right.

Problem No. 22  –  Solve for y:    9  -  6y  ≤  -45

                                                      -9                  -9       Subtract 9 or add -9

                                                            - 6y  ≤   -54

                                                             -6    ≤    -6        Divide by -6.

                                                                y   ≥    9

   Solution set: {y│y ≥ 9}

Graph on the number line, a closed circle at 9 and a shaded arrow to the right.

Problem No. 24 – Solve for y: -4 < y - 2 ≤ 10 This is called a conjunction. First step is to eliminate the constant (number) from the inside.

Then divide by the numerical coefficient of the variable (in the middle) if there is one.

               -4   <   y   -   2   ≤   10

                2                  2          2           Add to 2 in three places.

                -2   <        y      ≤   12            The y has a one in front of it, so this is

                                                              the answer.

      Solution set: {y│-2 < y ≤ 12}

Graph: On the number line, there is an open circle at -2 and a closed circle at 12 and the line is shaded between them.

Problem No. 28  –  Solve for k:    -15  <  8  -  4k  ≤  -8

                                                       -15  <  8  -  4k  ≤  -8

                                                         -8     -8               -8      Subtract 8 or add -8.

                                                        -23  <      - 4k  ≤  -16

                                                         -4             -4        -4       Divide by -4

                                                       23/4  >  k  ≥  4    By dividing by -4 the direction of the inequality was turned around. 

                4 ≤ k < 23/4     To be able to read the inequality, the smallest number should be on the left, so turn the whole conjunction around and the inequality signs change.

      Solution set: {k│4 ≤ k < 23/4}

Graph: On the number line, there is an open circle at 4 and a closed circle at 23/4 and the line is shaded between them.