Thursday, September 10, 2009

Why Study Math? - Solving Multi-Step Linear Inequalities - Part I

Unlike an equation, which shows balance or harmony among two expressions, an inequality shows imbalance. That is, an inequality states that one expression is bigger than or less than another expression. Linear inequalities involve expressions which are lines, or linear equations. These inequalities are solved much the same way as equations with one key difference. Read on to enter this curious world.

Let us take the linear equation 2x + 1 = 5. You know to solve for the unknown, x, you subtract 1 from both sides and then divide by 2. Thus x = 2. If we replace the "=" symbol by an inequality symbol such as "<", less than, or ">" greater than, we have a linear inequality. If we choose the former we have 2x + 1 < 5, which establishes the "imbalance" between the left and right hand sides of the expression. What this says mathematically is that the quantity 2x + 5 is strictly less than the quantity 5. (Note: we can also use the inequality symbols < and > with the = symbol to get <=, or less than or equal to, and >= greater than or equal to. In these cases, the "=" symbol permits the right and left sides of the expression to be equal.)

To solve an inequality, we carry out the exact same steps as if it were an equation, keeping the inequality symbol intact. Thus in 2x + 1 < 5, we subtract 1 from both sides and divide by 2 to get x < 2. Whereas equations place strict requirements on the value of the solution, an inequality permits a broad array. To be precise, 2x + 1 = 5 permits x to be only 2; in 2x + 1 < 5, however, x can be any value less than 2. If we allow x to be any kind of number---integral, rational, or irrational---we have infinitely many possibilities to choose x from.

You can visualize the inequality just solved as follows: picture a balance scale with a 5-weight on the right and a (2x + 1)-weight on the left. Since the inequality states that the 5-weight is bigger, the scale is tipping to the right. That is the left side is higher than the right side. Solving to get x < 2, tells us that in order for the "imbalance" to be maintained-for the right side to continue being lower than the left side-then x has to be any weight less than 2.

Take another example: 3y - 4 >= 8. In words, this inequality states that the quantity 3y - 4 is bigger than or equal to the quantity 8. Visually in terms of our balance scale, this inequality says that the left side is in balance with or bigger, that is tipping left, than the right side. To solve, we simply add 4 to both sides and then divide by 3. Doing so gives that y >= 4. Thus if we choose y to be any weight equal to or bigger than 4, our given "imbalance" or balance, in the case when both sides are equal, will be maintained.

The next time you see an expression involving one of the "imbalance" symbols, know that you have an inequality, and that to solve, really requires nothing new. There are some situations, however, that require a little more care. Those we get to in Part II.

Joe is a prolific writer of self-help and educational material and is the creator and author of over a dozen books and ebooks which have been read throughout the world. He is a former teacher of high school and college mathematics and has recently returned as a professor of mathematics at a local community college in New Jersey.



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