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## Algebra 1

### Course: Algebra 1>Unit 2

Lesson 5: Multi-step inequalities

# Inequalities with variables on both sides (with parentheses)

Sal solves the inequality 5x+7>3(x+1), draws the solution on a number line and checks a few values to verify the solution. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• say you have to graph an inequality,once you solve the equation such as :2r+5<19 would be 2 times 7 +5=19 right
but after that when you graph this on a # line how do you know which # to put the hollow or solid circle above? • If the it's just < or >, then you draw a hollow circle because your not including that point. x>4.You are not including 4 on the number line, but all the points greater than 4. x<3. You are not including 3 on the number line but all the points less than 3.

If the inequality sign is greater than or equal to or less than or equal to, then you shade the dot because your including that point. x is greater than OR equal to 4. X can be greater than 4 OR it can be equal to 4, so since 4 is one of the solutions, you need to use the solid dot. If x is less than or equal to 3, then you shade the dot because three is part of the solution set, x is greater than OR equal to 3.
• Do you also Swap The Symbol if you're ADDING or SUBTRACTING by a negative number? • I wish that flipping the >/< based on a negative or positive number being divided out was explained more. • in equation we do things on both side so its true. since inequations < are not equation= why we apply same rules • The rules are not exactly the same. The rules used maintain the relationship of the 2 sides of the inequality.

1) If we add/subtract the same value to both sides of an inequality, the relationship is unchanged. For example:
2<5 becomes 6<9 if we add 4 to both sides. The left side is still less then the right side.

2) If we multiply or divide both sides by the same positive value, the relationship is unchanged. For example:
3<9 becomes 6<18 if we multiply both sides by +2. The left side is still less than the right side.

3) This is the rule that is different. If we multiply or divide both sides by the same negative value, the relationship between the numbers reverses. So, we change the direction of the inequality. For example:
-2<7 becomes 4>-14 if we multiply both sides by -2. The left side is now larger than the right side, so we reverse the inequality.

Hope this helps.
• At couldn't you subtract 3 instead of 7? • when Sal divided the 2x and the -4 why did > not become < ? • When I do my math, my sign comes out flipped. Am I doing something wrong?

5x+7 > 3(x+1)
5x+7 > 3x+3
2x > -4
x < -2 • It should not be flipped. You only need to flip the sign when you multiply or divide both sides by a negative number. In that last step, you are dividing by 2 which is a positive number. So your sign should not be flipped. Hope this helps!
• I guess "false" and "no solution" are the very close, if not identical, and close also to "undefined" in meaning. E.g.:
Is "y = x/0" false? Is the system of equations "y = 3x and y = 3x+1" false? (I think so, since they can't both be true (for the same x). "4 < 3" seems to be just false, and for this, "no solution" seems inappropriate.

The system "2y = 2x+2 and 7y = 7x+7" is true for all x. I don't understand how "2 < 3" is true for all x when there is no x in the inequality. To me it's just a true statement about 2 and 3.

Does anyone have any thoughts about these things one way or the other? • Note: The following is from my own thought. You can agree or disagree with me.

`y=x/0` is not necessarily false. Dividing by 0 is undefined. It cannot be wrong should there be no right.

Now let's talk about `2<3`. Let's say you have an inequality and you manage to get to this point. Since 2 IS less than 3, that solution (or inequality without a variable) would be true.
For example, we can have `4<6+2`. Simplify that and you will get `1<2`, which proves the inequality is true.

I would also note that "no solution" and "false" have similar meanings. I will illustrate this.

In `4<3`, 4 is obviously not less than 3. I would, however, say it is "false", since there are no variables to make 3 greater than 4 or 4 less than 3.

However, for `y=3x` and `y=3x+1`, I would say there is "no solution", since (for the same x) there is no way to make the equations equal to each other.

My conclusion is that "false" and "no solution" have similar but not quite the same meanings. "Undefined" has a completely different meaning from "false" and a rather different meaning compared to "no solution."

• My sign comes out flipped. What am I doing wrong?

5x+7>3x+3 /-3
5x+4>3x /-5x
4>-2x /:-2
-2<x  