SOLUTION: when solving inequalities, how do you know whether to add or to subtract on both sides? Can you explain with an example?

Question 118192: when solving inequalities, how do you know whether to add or to subtract on both sides? Can you explain with an example?
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
You can work inequalities just as you would work and equation ... same rules:
If you add a number on one side of the inequality you must also add the same number to the
other side of the inequality.
.
If you subtract a number from one side of the inequality you must also subtract the same number
from the other side of the inequality.
.
If you multiply one side of an inequality by a number, you must also multiply the other side
of the inequality by the same number.
.
If you divide one side of an inequality by a number, you must also multiply the other side
of the inequality by the same number.
.
One exception you must remember ... if you multiply or divide both sides of an inequality
by a NEGATIVE quantity, then you must reverse the direction of the inequality sign each time
you multiply or divide by a negative.
.
And, just as you do with an equation, you solve it for the positive value of the variable.
.
Suppose you have the inequality 2x + 5 < 5x - 1 and you want to solve it. You can sort
of imagine that the inequality sign is an equal sign while you do the work.
.
First get rid of the + 5 on the left side by subtracting 5 from both sides. When you do that you
get:
.
2x < 5x - 6
.
Next get rid of the 5x on the right side by subtracting 5x from both sides, just as you would
do in an equation. After that subtraction from both sides you have:
.
-3x < -6
.
You can now solve for +x by dividing both sides by -3 to get:
.
x < -6/-3
.
BUT you divided both sides by -3 a negative number so you have to reverse the direction of
the inequality sign which results in:
.
x > -6/-3
.
Now just do the division on the right side and you get the answer:
.
x > 2
.
Not too bad, once you get used to the process. You can check your answer by selecting a
value of x that is greater than 2 and seeing if it works in the original inequality. The
original inequality was:
.
2x + 5 < 5x - 1
.
Let x be 3 and it becomes:
.
2(3) + 5 < 5(3) - 1
.
Which simplifies to:
.
6 + 5 < 15 - 1
.
Combine the numbers on both sides and you have:
.
11 < 14
.
That's true ... 11 is less than 14. So that seems to work. Next let x = 2 which means that
you have chosen a value for x that is NOT GREATER than 2. The original inequality is:
.
2x + 5 < 5x - 1
.
Substitute 2 for x and it becomes:
.
2(2) + 5 < 5(2) - 1
.
Do the multiplications and you get:
.
4 + 5 < 10 - 1
.
Combine the numbers on both sides and you have:
.
9 < 9
.
This is not true because 9 is not less than 9 ... it is equal to 9. It looks as if the answer
.
x > 2
.
is the correct solution to the problem.
.
Hope this gives you some help in working inequality problems. If you still need additional help,
re-post some more problems.
.
And Happy 2008 to you and yours ...
.
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