Algebra: Graphing Solutions
In Graphing Linear Equations, when I was discussing linear equations, I explained why it was important to draw graphs. Since linear equations contained two variables, usually x and y, there were an infinite number of ordered pairs that made each equation true. Because you couldn't write that infinite list of answers, it was useful to represent them with a drawing (graph) of the solutions.
You now know that basic inequalities have an infinite number of solutions, so you should also use graphs to help visualize their solutions. However, basic inequalities (like those you just learned to solve in the previous section) are different than linear equations because they only contain one variable. Therefore, you don't need a coordinate plane to graph basic inequalities; all you need is a number line, pictured in Figure 7.1; essentially, the number line is just the x-axis from the coordinate plane; since there is no second variable, you don't need a second axis on the graph.
Here's how to graph the solution to a basic inequality:
Make sure to get x on the left side of the inequality when you solve it. If you end up with 13 < x as a solution, you can reverse the sides, but if you do so, you also need to reverse the inequality sign like this: x > 13. If you forget to do so, the arrow on your graph will end up pointing the wrong way.
A solid dot on a number line graph indicates that the given number should be included as a possible solution, whereas an open dot indicates that the given number cannot be a solution. For example, if you graph x > 7, you place an open dot at 7 because it's not a valid answer (7 is not greater than itself).
- Solve the inequality. Before you can draw the graph of an inequality, you need to know its solution.
- Draw a number line. The number line doesn't always have to be centered at 0. It's sometimes more useful to center the number line at or near the value you found in the solution.
- Dot the hot spot. When you solve the inequality, you'll get something like this: x ≤ a or x > a, where a is a real number. If the inequality symbol allows for the possibility of equality (≤ or ≥), draw a solid dot at a. If, on the other hand, the symbol is either < or >, draw an "open dot," or a circle, at the a value.
- It's not impolite to point. Draw a dark arrow from your dot that points in the direction of the inequality symbol. If your final solution contained one of the less than symbols, draw the arrow to the left. If the solution contained a greater than symbol, the arrow (like the symbol) should point right.
Example 3: Graph the solution to 3(1 - 2x) < -5x + 6.
Solution: First you need to solve the inequality for x. Remember, you want to get x on the left side of the inequality, so once you distribute the 3, separate the variable by adding 5x to and subtracting 3 from both sides.
- 3 - 6x < -5x + 6
- 3 - x < 6
- -x < 3
Multiply (or divide) both sides by -1 in order to eliminate the negative sign in front of the x, and don't forget to reverse the inequality sign.
- x > -3
To graph this solution, you'll draw an open dot (since the inequality symbol does not contain "or equal to") at -3 and sketch a dark arrow from that dot that extends to the right (since the inequality sign, like an arrowhead, points to the right), as shown in Figure 7.2.
You've Got Problems
Problem 3: Graph the solution to 1 - x ≥ 2x - 5.
The graph makes it clear that every real number to the right of (but not including) -3 on the number line will make both the inequality x > -3 and therefore the original inequality, 3(1 - 2x) < -5x + 6, true.
Excerpted from The Complete Idiot's Guide to Algebra © 2004 by W. Michael Kelley. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.