Fill in the blank in the following rule on inequalities. If 𝑦 is greater than 𝑚, then 𝑦 plus 𝑎 blank 𝑚 plus 𝑎.
So we need to fill in that blank. So we will be filling in this blank with either an equal sign, a greater than sign or a greater than or equal to sign, or a less than or a less than or equal to sign.
Notice we already know that 𝑦 is greater than 𝑚. So looking at what we have, notice that there is an 𝑎 on both sides of the blank. So if we would take away an 𝑎 from both sides of this possible equation or inequality, which it would be an equation if we would use an equal sign and then it would be an inequality if we use an inequality sign, so if we would take away 𝑎 from both sides, the 𝑎s would cancel, and now we have 𝑦 blank 𝑚. Well we already know something about 𝑦 and 𝑚: 𝑦 is greater than 𝑚.
So we can use the greater than sign. So let’s think about this for a minute. Before we decide on the greater than sign, let’s think about an example. If 𝑦 is greater than 𝑚, we can let 𝑦 be eight and 𝑚 be five. Eight is greater than five; that is true.
So when we have 𝑦 plus 𝑎 is blank 𝑚 plus 𝑎, let’s plug in eight for 𝑦 and five for 𝑚. Now let’s choose a number for 𝑎. It could be any number. Let’s let it be ten. So we have eight plus ten on the left to give us eighteen and five plus ten on the right which is fifteen.
And eighteen is greater than fifteen, so no matter what we plug in for 𝑎, we’re still gonna have that the left-hand side is greater than the right-hand side. So our final answer will be 𝑦 plus 𝑎 is greater than 𝑚 plus 𝑎.