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Equations and Inequalities

Weve talked a little bit about equations so far, but now, ‘we want to give some definition to the term. An equation consists of two mathematical expressions separated by an equal sign.

So, for instance, we might have an equation that says 1+1 = 2, and this is true, because this value is the same as that value.

Mini-test: Equations and Inequalities 

Suppose the average boy's height grows 7cm per year starting from birth at 40cm.  Which equation represents greater than average growth?
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B.
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Suppose the average worker's income grows $2k per year starting from $25k annual salary.  Which equation represents income lower than average?
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B.
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Suppose the average worker's production costs 2 hours per part plus 45 minutes to clean up at the shift's end.  Which equation represents better-than-average productivity?
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B.
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Which inequality is entirely outside the area between lines:  y = x + 1  -and-  y = 21 - 3x  for 0 < x < 5?
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Which inequality can reach the area between lines y = 2x + 4, y = 24 - 3x for 0 < x < 4?
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B.
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E.

 

Next Lesson: Graphing Solutions to Linear Inequalities

The following transcript is provided for your convenience.
Now, the beauty of equations is that you can perform any operations the same way on both sides of the equation, and the equation remains true. So, for instance, we can add 5 to both sides of this equation, and the equation is still true.

We can multiply both sides of the equation by 7, and the equation is still true.

We can divide both sides of the equation by 13, and this equation is still true.

And so, anything you perform on both sides of the equation equally do not invalidate the equation.

Similar to equations, we have what’s called an inequality. So, we might have the inequality 1 < 2. And so, this consists of a mathematical expression on one side, and a mathematical expression on the other side, and a sign indicates which side is greater or lesser.

So, that’s what we have here. This is read as 1 is less than 2. And, similar to equations, we can perform the same operations on both sides of the inequality sign, and it can remain true.

So, again, we can add 5 to both sides, and it’s still true, 6 is still less than 7.

We can multiply both sides by 7, and it’s still true. This side is now 42, and this one is equal to 49, so it’s still true.

And we could divide by 13, and it’s still true.

The one thing that we can’t do with inequalities without changing what we can with equations is multiplying or dividing by a negative number.

So, if we wanted to multiply both sides of this inequality by -1, that would make it untrue, because in the most simplest of examples, if we have an equation that says 1 < 2, and we multiply both sides by -1, that becomes -1 < -2, which is not true.

So, what you have to do, if you multiply or divide both sides of an inequality by a negative number, what you have to do is reverse the inequality sign. And so, this is now a valid inequality once again, because we switched this sign after multiplying by -1.

To show in real simple terms what that looks like, if you have 1 < 2, and then you want to make it negative, you can say -1 > -2, and that is the true expression of the inequality.

And so, you can have greater than, or less than, and you can also have a greater than or equal to, which is something of a combination of equations and inequalities. So, we would write greater than or equal to as a greater than sign with a line under it. And similarly, less than or equal to would be less than with a line under it.

And so, this is kind of a more formal definition of equations and inequalities, and what you can do with them.

The next lesson: Graphing Solutions to Linear Inequalities.