Special Cases

Equations with Infinite Solutions and Equations with No Solution

[Video begins with a white screen with the category ‘Infinite Solutions, (All Real Numbers)’ on the left, and the category ‘No Solution’ on the right. There is a yellow box in the center of the screen that says, “Special Cases” at the top, followed by “Equations with Infinite Solutions” underneath it, and finally “Equations with No Solution” below that.]

Narrator: Hello! In this video we’re going to talk about equations that fall into two special cases: ones that have infinite solutions and ones that don’t have any solution at all. First, let’s give an example of one that has infinite solutions. [The yellow box disappears and the equation “‘2x + 3 = 2x + 3” appears on the screen on the ‘Infinite Solutions’ category.] Here’s an example of an equation: 2x plus 3 equals 2x plus 3. Now, let’s solve for x. The first thing that we’re going to do is add a negative 3 to both sides [“+-3” is written underneath both sides of the equation]. Now let’s rewrite it. 2x equals 2x [equation is written on the screen as the narrator says it]. Now, we can isolate x by multiplying by the multiplicative inverse. [Both instances of 2x are placed in parenthesis, with ½ on the outside of both parenthesis.] So one-half on the left and multiply by one-half on the right.

So now we have x equals x [writes “x = x” on the screen]. Well this is always true, no matter what value X is. So in this case, X can be any real number or, in fact, the solution to this is all real numbers. [Screen returns to original equation (2x + 3 = 2x + 3).]

Now when we solved this the first time, the first thing we did was add the inverse of 3. This time, however, let’s start by adding the inverse of 2x to get the X’s all on the same side. [“+-2x” is written underneath both sides of the equation.] So if we add a negative 2x to both sides, we get 2x plus negative 2x is zero, plus three equals 2x plus negative 2x is zero plus 3. [“0 + 3 = 0 + 3” is written on the screen.] So three equals three. [“3 = 3” is written.] And again, this is always true, no matter what value of X we had put into this equation. So this shows again that the solution to this equation is all real numbers.

Anytime you solve an equation and you get something like 3 equals 3 or X equals X [“3 = 3” and “x = x” are written at the bottom of the screen], it means there’s an infinite number of solutions. In fact, all real numbers are a solution.

Now, let’s look at this other equation: 2x plus one equals 2x minus five [2x + 1 = 2x - 5” appears on the right side of the screen under the ‘No Solution’ category]. Let’s start by adding the inverse of the 2x to combine the X’s on the same side. So we’re going to plus a negative 2x to both sides [Writes “+-2x” under each side of the equation]. Well 2x plus negative 2x on the left-hand side is zero, and we’re left with one, and 2x plus negative 2x on the right-hand side is also zero, so we’re left on the right-hand side with just negative five [Writes 1 = -5 on the screen]. But this is a false statement. One is not equal to negative five, and since one is not equal to negative five, no matter what value we put into this equation for X, it won’t ever be true [A red slash is drawn over the equals sign].

[The right side of the screen is reset to the original equation, “2x + 1 = 2x - 5.”]

This time, let’s try combining the like terms of the terms that don’t have a variable. So let’s add a negative one to both sides. [Writes “+ -1” under both sides of the equation] Plus negative one, plus negative one. So now we’re left with 2x—one plus negative one is zero on the left-hand side— so we just have 2x on the left-hand side [writes “2x” under the left side of the equation], and now we have 2x minus five, plus a negative one on the right-hand side. Well, negative five plus negative one is negative six. 2x minus six [writes “2x - 6” under the right side of the equation]. Once again, our next step will be to combine the terms with a variable. So we add negative 2x to both sides [writes “+ -2x” on both sides of the equation]. We’re trying to get rid of the 2x on the right-hand side, and combine it with the one on the left-hand side. But on the left-hand side 2x plus negative 2x is still zero, which equals 2x plus negative 2x minus six—well that’s equal to zero minus six [writes 0 = -6 on the screen]. And again, this is a false statement. Zero is not equal to negative six.

Anytime you solve an equation and come up with a false statement at the end, it means that there are no solutions [underlines “No Solution” up at the top of the screen in red]. There is no value of X that will ever make zero equal negative six [a red slash is drawn over the equals sign].

So let’s just go over this one more time. Anytime you solve an equation and get the same thing on each side of the equal sign, such as three equals three or X equals X, all real numbers are the solution [underlines “All Real Numbers” in green up in the top-left corner of the screen]. Anytime you get a false statement, such as zero equals negative six, there are no solutions to that equation.

[End of video.]