## Video and Examples

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So far all the equations we have solved involved positive numbers. In this lesson, we learn how to model negative numbers with bags and marbles. We will differentiate positives and negatives with different colors.

 Positives (green) Negatives (red) $x + 4$ $-x + (-4)$ $3x + 6$ $-2x + (-5)$

Expressions can include a mix of both positives and negatives. Here are some examples of how to model such expressions.

 Expression Model $-2x + 4$ $3x + (-5)$ In earlier lessons, we learned that the strategy for finding the number of marbles in each bag was to move bags and marbles from the scale until there was only a bag on one side and marbles on the other. We will continue to use this strategy, but we will have to use a new way of thinking for how to ÒremoveÓ things.

 To remove a positive bag you need to add a negative bag. To remove a negative bag you need to add a positive bag. To remove a positive marble you need to add a negative marble. To remove a negative marble you need to add a positive marble. Example 1 How many marbles are in each bag? To find the number of marbles in each bag we need to add positive and negatives bags and marbles so that only one positive bag remains on one side and all the marbles (positives or negatives) are on the other side.

 Model Algebraic equation Verbal  The equation  Added 2 positive marbles to both sides.  Added 1 positive bag to both sides.  Divided both sides by 3.  The solution. There are two positive marbles in each bag.

 Example 2 How many marbles are in each bag?

 Model Algebraic equation Verbal  The equation  Add 2 positive marbles to each side.  Add 3 negative bags to each side.  Divide both sides by 2.  The solution.

 Example 3 How many marbles are in each bag? NOTE: It is assumed that we are looking for the number of marbles in one positive bag of marbles.

# Self-Check

 Question 1 Solve for x. $-x+(-5) = 2x+1$ [show answer]   $\matrix{-x+(-5)&=&2x+1\\+x&&+x\\&&\\-5&=&3x+1\\+(-1)&&+(-1)\\&&\\-6&=&3x\\\overline{3}&&\overline{3}\\&&\\-2&=&x}$

 Question 2 Solve for x. $-2x + (-4) = 3x+1$ [show answer]   $\matrix{-2x+(-4)&=&3x+1\\+4&&+4\\&&\\-2x&=&3x+5\\+(-3x)&&+(-3x)\\&&\\-5x&=&5\\\overline{-5}&&\overline{-5}\\&&\\x&=&-1}$

 Question 3 Solve for x. $-x+6 = x+(-2)$ [show answer]   $\matrix{-x+6&=&x+(-2)\\+x&&+x\\&&\\6&=&2x+(-2)\\+2&&+2\\&&\\8&=&2x\\\overline{2}&&\overline{2}\\4&=&x}$