If in your algebra course you were asked to represent inequalities in a graph, this article can help you. Inequalities can be represented on a line of real numbers or on a coordinate plane (with the x and y axes): both of these methods are good representations of an inequality. Both methods are described below.
Steps
Method 1 of 2: Method of the line of real numbers
Step 1. Simplify the inequality you need to represent
Multiply everything in parentheses and combine the numbers that are associated with the variables.
-2x2 + 5x <-6 (x + 1)
-2x2 + 5x <-6x - 6
Step 2. Move all terms to the same side, so that the other side is zero
It will be easier if the variable at the highest power is positive. Combine common terms (for example, -6x and -5x).
0 <2x2 -6x - 5x - 6
0 <2x2 -11x - 6
Step 3. Solve for variables
Treat the sign of inequality as if it were an equal and find all the values of the variables. If necessary, resolve with common factor recollection.
0 = 2x2 -11x - 60 = (2x + 1) (x - 6) 2x + 1 = 0, x - 6 = 02x = -1, x = 6x = -1/2, x = 6
Step 4. Draw a line of numbers that includes the solutions of the variable (in ascending order)
Step 5. Draw a circle over those points
If the inequality symbol is "less than" (), draw an empty circle over the solutions of the variable. If the symbol indicates "less than or equal to" (≤) or "greater than or equal to" (≥), then it colors the circle. In our example the equation is greater than zero, so use empty circles.
Step 6. Check the results
Choose a number within the resulting ranges and enter it into the inequality. If, once solved, you get a true statement, shade this region of the line.
In the interval (-∞, -1/2) we take -1 and insert it into the initial inequality.
0 <2x2 -11x - 6
0 < 2(-1)2 -11(-1) - 6
0 < 2(1) + 11 - 6
0 < 7
Zero less than 7 is correct, so shade (-∞, -1/2) on the line.
In the interval (-1/2, 6) we will use zero.
0 < 2(0)2 -11(0) - 6
0 < 0 + 0 - 6
0 < -6
Zero is not less than six negative, so don't shade (-1/2, 6).
Finally, we take 10 from the interval (6, ∞).
0 < 2(10)2 - 11 (10) + 60 <2 (100) - 110 + 60 <200 - 110 + 60 <96 Zero less than 96 is correct, so shade (6, ∞) Use arrows at the end of the shaded area to indicate that the interval continues indefinitely. The number line is complete:
Method 2 of 2: Coordinate plane method
If you are able to draw a line, you can represent a linear inequality. Simply think of it as any linear equation in the format y = mx + b
Step 1. Solve the inequality according to y
Transform the inequality so that y is isolated and positive. Remember that if y changes from negative to positive, you will have to flip the inequality sign (greater becomes smaller and vice versa). Y - x ≤ 2y ≤ x + 2
Step 2. Treat the inequality sign as if it were the equal sign and represent the line in a graph
USA y = mx + b, where b is the y intercept and m is the slope.
Decide whether to use a dotted or solid line. If the inequality is "less than or equal to" or "greater than or equal to", use a solid line. For "less than" or "greater than", use a dashed line
Step 3. Consider shading
The direction of the inequality will determine where to shade. In our example, y is less than or equal to the line. It then shades the area below the line. (If it was greater than or equal to the line, you should have shaded above the line).
Advice
- First, always simplify the equation.
-
If the inequality is less than / greater than or equal to:
- use colored circles for a number line.
- use a solid line in a coordinate system.
-
If the inequality is less than or greater than:
- use unstained circles for a number line.
- uses a dashed line in a coordinate system.
- If you can't solve it, enter the inequality in a graphing calculator and try working in reverse.
