How to Solve Two Step Inequalities
In this lesson, let’s learn how to solve two-step inequalities. We are given two problems, we need to solve them both and graph them as well on the number line. Let’s do them one by one. When we are looking at inequalities, we see that the expression looks very similar to the equation except that instead of the equal to sign, we’ve got a greater than equal to sign or a greater than sign or less than sign or less than equal to sign.
Now, we solve them exactly the way we would solve equalities except for one change. In order to undo the operations or inverse the operations, if we multiply or divide the left and the right inside by a negative number, if we multiply or divide by a negative number then we need to reverse the signs, reverse the inequality. Let’s use them in practice. So, let’s solve Part A. Part A gives us (-2x+3≥8). I’m using a cursive x or a curved x so we don’t confuse it with the multiplication sign so, (-2x+3≥8). As we can see, the variable which is x, I have added three to it. So, the first thing we do is reverse the operation by subtracting three on both sides. So on the left side, we get (-2x), the inequality comes down and (8-3) is 5. So, if (-2xis≥5), as we can see the variable x has been multiplied by (-2) so we’ll divide by (-2). We’ll divide by (-2) on both sides. So here, I got (-5/2) and here I get x. Because we divided by a negative number, we have to reverse the inequality, so greater than equal to sign became a less than equal to sign. So, the solution that we’re looking for is (x≤-5/2), (-5/2) is the same as (-2/5) or 2.5.
So, let’s draw the number line. Let’s draw 0, 0.5, -0.5, -1, -1.5, -2, -2.5, 3 etcetera. We just draw a number line. When we are looking for values of (x≤-2.5), (-2.5) is right here. So, the answer that we’re looking is any value that is less than (-2.5) or equal to (-2.5). That’s the reason we have a close or a filled in circle at (-2.5) because that value of x being equal to (-2.5) is also included in the solution set. So, that’s the number line that represents this particular solution.
Lets’ switch and try this second part which is (t/-3>-12).As we can see, the variable t has been divided by (-3) so let’s multiply by (-3) on both sides. (-12x-3) is 36, (t/-3x-3) gives us t but here since the sign is greater than we will reverse it to get (t<36). If we divide or multiply by a negative number, the sign, the inequality gets reversed. So again, drawing the number line, we can do (0, 12, 24, 36, and 48). Here, I can have -12,-24, etcetera and since (t<36) this is the solution set that we’re looking at, (t<36) and there’s an open circle at 36 because there is no equal to sign.
So, what are the things that we’ve learned in this particular lesson? If we are solving inequalities, we solve them pretty much the same way as we solve equalities or equations, except if we multiply or divide by a negative number. We have to remember to reverse the inequality like we reverse the inequality here.
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