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Solving and Graphing Multistep Inequalities - Problem 2

Teacher/Instructor Alissa Fong
Alissa Fong

MA, Stanford University
Teaching in the San Francisco Bay Area

Alissa is currently a teacher in the San Francisco Bay Area and Brightstorm users love her clear, concise explanations of tough concepts

In multi-step inequalities, follow the same general rules as solving equations. Start by simplifying both sides of the inequality sign, as needed. This means to get rid of any parenthesis by distributing, combine any like terms, and so forth. When using the distributive property, it is important to distribute the sign of the number as well. After distributing, combine like terms. Once the inequality has been simplified, use inverse operations to solve for the unknown variable. Remember that solving an inequality and finding all the values of the variable requires "undoing" what has been done to the variable, working in the reverse order of PEMDAS. Remember to change the inequality sign to the other direction if both sides are being multiplied or divided by a negative number. When graphing, keep in mind when to use an open circle versus a closed circle, and which direction the arrow should be pointing.

I can tell this inequality is going to involve multiple steps, because there is x's on both is sides of inequality sign. Also because there is some distributing that's going to have to happen. I'm going to have to distribute, combine like terms, get all my x's together, all my constants together. Then along the way, I might have to change the direction of the inequality. Let's try it.

The first thing I'm going to do is simplify each expression, each side of the inequality. Distribute that 3, so I'll have 3x take away 9. Now I'm going to distribute the -2. Be careful not to lose that negative sign. Negative times negative gives you a +2x right there. All I've done so far is simplify.

Next thing I'm going to do now that it's totally simplified is to start solving. Solving is where you add stuff to both sides of the equation. So like for example, if I want my constants to be together, I'm just going to choose to add 9. I could also add 2, it would still give me the same result. But I'm just going to choose to add 9. So now I'll have 3x is less than or equal to 7 plus 2x. I'm getting there.

My x's are still on opposite sides of the inequality. So I need to move this one over there by subtracting 2x's from both sides. Now I'll have x is less than or equal to 7. I'm going to write my 7 with that slashy thing, so that it's clearly a 7, and not like inequality sign. It's kind of confusing sometimes.

Once I have that, I need to graph the solution. So I'll draw my number line. There is 7. I want to mark a closed circle at 7, because x is less than or equal to 7. 7 is a solution to this inequality. I want to mark numbers that are less than 7, so my arrow is going to go out here towards the smaller values. This arrow continues on towards infinity. So this graph represents every single solution. Any number I were to pick in this area of my graph would be a solution to this original inequality.

The thing I want you guys to remember when you're solving this, is to treat it like an equal sign. Do the same kind of solving techniques you would if it were an equals; in terms of getting your x's together, simplifying each expression. Just make sure you're really careful if there is any dividing going on that you change the direction of the inequality.

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