Graphing inequalities

Linear inequalities with one variable

  1. Solve the inequality
  2. Represent the solutions on the real line.

Example: Solve and graph the inequality 1 ≤ x + 5 ≤ 3

To eliminate the (+5) that accompanies the x, we add (-5) to the three members of the inequalities, like this:

1 ≤ x + 5 ≤ 3 → 1 + (-5) ≤ x + 5 + (-5) ≤ 3 + (-5) → -4 ≤ x ≤ -2

The last expression indicates that x has a value between -4 and -2, both included, which we write down as: x = [-4, -2]

Graphing inequalities

 

Linear inequality system

  1. Transform inequality into equality
  2. Graph the line. For this, we give one of the two variables two values, with which we obtain two points; by representing and joining these two points we obtain a line. If the inequality is <or>, the line is a dashed line. If the inequality is ≤ or ≥, the line is a continuous line.
  3. Take a point and replace it on the inequality. If it is fulfilled, the solution is the region where it is located, otherwise the solution will be the other region.

Example: Solve and graph 2x + y ≤ 1

  1. 2x + y = 1
  2. y = -2x + 1; To graph, we give one of the two variables two values, with which we obtain two points:

When x = 0; y = -2 (0) + 1 = 1

When x = 1; y = -2 (1) + 1 = -1

By representing and joining these two points (0,1) and (1, -1), we obtain a line

Graphing inequalities cover

  1. Let's take the point (0,0).

2x + y ≤ 1 → 2 (0) + (0) ≤ 1 → 0 ≤ 1 False

Therefore, the points of half plane 1 are part of the solution.

Linear inequalities with a variable with absolute value

The same steps are followed to graph linear inequalities with one variable.

Example: graph the inequality | 2x - 2 | > 2

| 2x - 2 | > 2 → 2 <2x - 2 <-2 → 2 + (2) <2x - 2 + (2) <-2 + (2)

4 <2x <0 → 4/2 <2x / 2 <0/2 → 2 <x <0

The last expression indicates that x is greater than 2, that is x = (2, + ∞), and that x is less than 0, that is x = (-∞, 0). The answer is the union of these two intervals.

Graphing inequalities example

 x = (-∞, 0) ∪ (2, + ∞)