graph the inequality

Question

Solve each inequality.Graph the solution.

|z-3|+2≥4

Answer 1

Given inequality is

Z belongs to [1 5]

Comments

The solution set of | z - 3 | + 2 ≥ 4 is {z Є R | z ≤ 1 or z ≥ 5}.

Answer 2

The Absolute Value Inequality is | z - 3 | + 2 ≥ 4

Subtract 2 from each side.

Then, the inequality is | z - 3 |  ≥ 2.

• First we consider the case, where z - 3 ≥ 0, i.e. z ≥ 3. In this case |z - 3| = z - 3, so we can write the inequality as z - 3 ≥ 2.

Solve the inequality : z - 3 ≥ 2 for z.

z ≥ 2 + 3

z ≥ 5.

• Now consider the case, where z - 3 < 0, i.e. z < 3. In this case |z - 3| = - (z - 3) = 3 - z, so we can write the inequality as 3 - z ≥ 2.

- z ≥ 2 - 3
- z ≥ - 1

z ≤ 1.

So a real number z is a solution of the original inequality if z ≥ 3 and z ≥ 5

                                                                                      or if z < 3 and z ≤ 1.

Thus, the solution set is {z Є R | z ≤ 1 or z ≥ 5}.

Graph of the solution set on a number line :

Observe the graph , the closed circle means that 1 and 5 are the solutions of the inequality.