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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
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
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
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
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).