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Adding or multiplying by a nonzero constant on both sides of an equation doesn’t affect the equality. Does the same fact hold for inequalities? In this lesson you will study this question for linear inequalities.
Below are the graphs of $y≥x$ and $y+2≥x+6$.
On the grid to the left, you are looking at the graphs of both of these inequalities on the same grid. The dark blue region is the overlapping region of the two inequalities. The pink region is the rest of the graph of $y≥x$.
Here are the graphs of $y≥x$ and $y+a≥x+a$. Remember that two inequalities are equivalent if they overlap completely (the shaded region is dark blue).
So far you have seen that adding the same constant to both sides of an inequality doesn’t change the graph of the inequality. Now, let’s look at multiplying both sides of an inequality by a constant.
Use the slider to change $a$ and try to find a value for $a$ that makes the inequality $y≥x$ equivalent to $-2y≥ax$.
In multByNeg, you saw that if you multiply both sides of an inequality by a negative number, you change the graph of the inequality. The question we are trying to answer is: "How can you multiply an inequality by a negative number and get an equivalent inequality?"
In multByNeg, you weren’t able to find any $a$ value that would make the inequalities equivalent.
If one side of an inequality is multiplied by a negative number, to find an equivalent inequality the “direction” of the inequality has to change (for example, $≥$ has to change to $≤$).
You are looking at the graphs of $y≥x$ and $ay≥ax$. Remember that two inequalities are equivalent if their graphs overlap completely (the shaded region is dark blue).