Solving Inequalities with a Balance Scale AnalogyDate: 08/31/98 at 16:07:50 From: Debbi Bross Subject: Algebra 2 Compound Sentences wuth Inequalities We have to graph: 0.3 - 0.7z > 0.4z - 3.0 and 4(z - 2) - 6 ≤ 15 Date: 09/01/98 at 13:00:12 From: Doctor Peterson Subject: Re: Algebra 2 Compound Sentences wuth Inequalities Hi, Debbi. Inequalities really aren't much different from equations, but you do have to be a little more careful. If you've seen equations compared to balanced scales, where the left side balances the right side, you can picture an inequality as an unbalanced scale, tipped to one side or the other. You can do most of the same things you can do to an equation, such as adding the same amount to both sides, and the scale will still remain unbalanced in the same direction. The only thing you have to watch out for is that when you multiply or divide by a negative number, the inequality switches over. For that reason, I generally avoid multiplying by a negative, and just work around that if I need to. Here's a picture of the scale flipping over when I multiply by -1. It's like reversing gravity so that what went up (a negative "weight," maybe a helium baloon, on each side) now comes down, because what was more negative (that is, less) is now more positive (that is, bigger): | -3 | 1 \ | / \ | / \ | / A \ -1 | 3 / A \ | / | -3 < -1 | 3 > 1 Let's do your first problem: 0.3 - 0.7z > 0.4z - 3.0 First we want to gather the z's on one side. We can do that by subtracting 0.4z from both sides, but then we would get a negative coefficient of z, which I'd rather avoid. I'll go ahead and do that anyway, so you can see what to do when that happens. Ordinarily I'd move the z to the right instead, but if you don't notice that, things will still work out. 0.3 - 1.1z > -3.0 Now we can get the non-z stuff out of the left side by subtracting 0.3 from both sides: -1.1z > -3.3 Now here's where the one bit of magic comes in. Let's just multiply everything by -1. Note that the inequality switches direction, as I explained above: 1.1z < 3.3 Now we can divide each side by 1.1 (which is safely positive): z < 3 Now to graph this, we just find the place where z = 3, and mark all values to the left of that: <===+===+===+===+===+===o---+---+---> -2 -1 0 1 2 3 4 5 I used an "open circle" at 3 because z = 3 does not satisfy z > 3. Now see if you can do the second problem. It's easier than this one. - Doctor Peterson, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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