How to Solve Inequalities ?
Now that you have learned so much about solving equations, you're ready to solve inequalities.
You might think that since an equation looks like this: x - 4 = 10 an inequality would look like this: x - 4 ≠10
In the formal sense, x - 4 ≠10 is an inequality.
In algebra, inequalities look exactly like equations, but instead of ‘=' (or ‘≠' ), you'll see:
< less than > greater than
≤less than or equal ≥greater than or equal
Let's solve the inequality x - 4 < 10.
To do so, consider how we solve x - 4 = 10.
We can use the same steps to isolate x and solve the inequality x - 4 < 10.
x - 4 < 10
x - 4 + 4 < 10 + 4
x < 14
Try different values for x. See if all values less than 14 make the inequality a true statement. Then, check if values greater than or equal to 14 make the inequality false. Finally, replace ‘<' with ‘≤ ' and solve x - 4 ≤10. How does the solution to x - 4 < 10 differ from the solution to x - 4 ≤10?
Now see if you can solve these inequalities: 2x + 4 > -8 and 2x + 4 ≥ -8
By adding the same number to both sides or subtracting the same number from both sides of an inequality, you do not change the balance of the inequality in any way.
Just to make sure this is so, look at the examples below and then make up some of your own until you're satisfied.
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-4 < -3
-4 + 4 < -3 + 4
0 < 1
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y - 5 ≤1
y - 5 + 6 ≤ + 6
y + 1 ≤7
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1 ≥-2
1 - 2 ≥-2 - 2
-1 ≥-4
|
Multiplication and division
For equations, you can multiply both sides by any non-zero number, positive or negative, without changing the equality.
This is not necessarily so for inequalities.
See if you can find a pattern in the following examples.
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-4 < 4
-4 /(-2) < 4 /(-2)
2 < -2
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-y ≤1
-y *(-1) ≤1 *(-1)
y *-1
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1 ≥-2
-5 *1 ≥-5 *(-2)
-5 *10
|
|
-4 < 4
-4 /4 < 4 /4
-1 < 1
|
y ≤12
y *2 ≤12 *2
y ≤24
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1 ≥-2
5 *1 ≥5 *(-2)
5 ≥-10
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What's happening here? In each case, we perform exactly the same operation on both sides of the inequality, just as we do with equations. However, in some cases, the result is incorrect. The results of the examples in the bottom row of the table are all valid. In these cases, we have multiplied or divided by a positive quantity. The results in the top row are all invalid. In these cases, we have multiplied or divided by a negative quantity.
Why do you think it makes a difference whether you multiply by a negative rather than a positive number?
When you multiply or divide a quantity by a negative number, you change its sign. When you multiply or divide both sides of an inequality by a negative number, the direction of the inequality changes.