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Solving and Graphing Multistep Inequalities - Problem 1
To solve a multi-step inequality, apply the same rules as solving an equation -- use inverse operations in the reverse order of operations (PEMDAS) to undo what is being done to the variable. Remember to change the inequality sign to the opposite direction if both sides are multiplied by a negative value. To graph on a number line, keep in mind a few rules: If the inequality is "equal to" a number, put a closed circle above that number. If the inequality is "greater than" or "less than" but not "equal to", then put an open circle above that number. Use an arrow to indicate all the values that fall within the inequality -- "greater than" or "greater than or equal to" means the arrow will indicate all the values above the number (to the right); "less than" or "less than or equal to" means the arrow will indicate all the values below the number (to the left).
This inequality problem is going to involve multiple steps. I can tell, because the variable m is both being multiplied by 2, and then you also have to deal with this subtracting 1 part. So I'm going to solve it just like I would solve if there was an equal sign.
Let's go ahead and check it out. Add 1 to both sides to isolate this 2 times m. Now I'm going to be dividing by +2. It's okay, nothing changes when I'm dividing by a positive value with inequalities. If I were dividing by a negative value, I'd have to change the sign. But since I'm just dividing by +2, those 2's cancel out. I'll be left with m is greater than 9. M can be any number bigger that 9. It could be 9.1, 9.10. 9.10 is not a real number, I just made that up. It could be 20, it could be 100, whatever. M could could be any number that's bigger than 9.
So if I put 9 on my number line, I want to make sure it's an open circle. And then, I'm going to mark my arrow out to the right to show numbers that are larger than 9. You guys can see this is not too difficult. It's a lot like solving equations. The only time you have to pause and check yourself, is when you get to this multiplying or dividing step. Since I divided by a positive value of 2, I didn't have to change the inequality direction. It was one of the easier ones. We'll see some more challenging ones as you guys progress through your homework.