Solving Algebraic Inequalities: Methods and Applications

Algebra With Inequalities Now we get a little bit more technical. Nothing too technical and show you how to do some algebraic manipulation with inequalities. We'll start by reminding you how to do it with equalities. In a nutshell, an inequality is a statement that two expressions are not equal to each other. An example is the statement "2x + 3 > 10". First, let's examine what you do mechanically and why you would do it. Then, we'll show you some new stuff—how to do it with inequalities and how to multiply by negative numbers in inequalities. And warn you that multiplying by a negative number is different from multiplying by a negative number in an equality. Let's get into it. First, we must recognize that the material you are about to learn was formerly taught using algebraic methods. If we start with something very simple, like "4=4," we see that no one truly appreciates what a powerful tool this simple statement is in an algebraic equation. What is said - that if you take an equality ("4=4") and add the same number to both sides ("4+3"), this will always yield another true statement. That is: "If 4=4 then 3=3." In fact, this statement can be simplified to 7=7 or "A+C" equals "B+C" or "A, B, C can be any integers." For example, suppose someone tells you there's a mystery symbol called X. And X plus three equals ten and they demand that you solve for X. Meaning, what they're really asking you is, what is X? If X plus three equals ten. You can solve this algebraically by first isolating x so that it is equal to something; then you would know how to proceed with your calculations. By subtracting three from both sides of the equation—in other words, writing X plus three minus 3 and then subtracting that from both sides—you will get X minus 3/10 on one side and 10 minus 3/10 on the other side. Now you can divide each side by 10 to discover that x equals 7/10. Let's start again with four equals four. And, let's multiply it by say two. Two times four equals two times four. This is eight. So, we can multiply the number. We can also multiply by a negative number. If we start with four equals four, and then multiply by negative three, the result is negative 12. Over here, this gives us negative 12, as does this.General rule, if A b and C are numbers, and C is not zero. And A equals b, then C times A equals C times b. So, suppose someone comes along and says minus five X equals 15. Here, you solve for X. If we multiply -3 by -5, we get 15. If we isolate the X, we can find that X equals -3. Another way to do it is to multiply both sides by the common denominator. We can use the principles of algebra to solve for unknown variables. For example, let's solve for X in the equation X=15-5(1/5). First, we can add five to both sides of the equation to get X=20. Next, we multiply both sides by negative one fifth to get -3X=-15 over five. This gives us X=3. Let us consider the inequality 4 < 7. If we add something to both sides, then 4 < 7 and 4 + 2 < 7 + 2. This is true because 4 +2 = 6 and 7 + 2 = 9. If we subtract something from both sides, then 4 < 7 and 4 - 1 < 7 - 1. This is also true because 4 1 = 3 and 7 - 1 = 6. So, general rule. If A is less than b then A plus C is less than b plus C. For any written numbers A, B and C we can even picture that on the number line. Right, suppose here is A, here's b. That means, A is less than b is being left of it. And I add C, if C is positive that means I'm shifting A by C and I'm shifting b by C. There's no way that A plus C can jump over b plus C. Likewise, if C is negative. We can use the method of solving algebraic inequalities to solve this problem. Suppose someone tells us that X plus three is less than 10. And they ask us to tell them more about what that means about X. So, if X plus three is less than 10, we can subtract three from both sides. And you all should practice with that. So let's just see. We subtract three from the left and we get X. I subtract three from the right and we get 10 minus three. In other words X is less than seven. So what does that mean? That means whatever on earth x is, it has to be to the left of seven. In other words, X has to live in this blue zone here. Any number wants to be as long as that doesn't exceed seven. And that's what the information that X plus three is less than 10 looks like. Now, here's a bit of advice. When you multiply by both sides of an inequality, it can be more dangerous. Let's start again with five less than eight. First, let's multiply by a positive number: three times five is 15; eight times five is 40. This works because three times five equals 15; eight times five equals 40, so 15 is less than 40. Let's take five less than eight and multiply by a negative number (–1). If we put minus one times five, the result is –5; if we put minus one times eight, I get –8. So what? Well, if you put minus one times both sides of an inequality and then solve it—if you do that every time—hoorah! You will get an answer that does not satisfy the original inequality. If we ever try to write this out, here is the number line: zero, five, and eight. Unfortunately, minus five is not before minus eight on the number line. In fact, minus five is greater than minus eight. Here's the general rule. Suppose, A is less than b. If C is greater than zero. A times C is less than b times C. But, if C is less than zero. Then A times C is greater than b times C. Suppose we are told that minus two times X is less than 10. What does this mean about X? Well, let's multiply both sides of the inequality by –1/2. If we multiply this side by –1/2 and minus 2X, we will multiply this side by –1/2 times –1/2 times 10. But now we have to flip the inequality because I multiplied both sides of the inequality by a negative number. This gives us X > –5 which pictured on the real number line here is 0, here is –5; thus X can be anything in here. So if X is anything in here and we feed X into this equation up here.