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Question 284835: How do I solve problems like this; 1/6(3x+10>5/12(x-1)? I'm having a hard time understanding these types of problems
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! You solve these the same way you would solve equality equations.
The only difference is that when you multiply both sides of the equation by a negative number, the inequality reverses.
Example:
6 > 5
Multiply both sides of this equation by -1 to get:
-6 < -5
The equality reversed.
If you had multiplied:
6 > 5 by -10, the equality would still have reversed.
You would have gotten:
-60 < -50
Other than that, you treat them the same way you would treat an equality equation.
Your problem is:
(1/6)*(3x+10 > (5/12)*(x-1)
Multiply both sides of this equation by 12.
This will remove the fractions on each side of the equation.
You will get:
2*(3x+10) > 5*(x-1)
Simplify by removing the parentheses to get:
6x + 20 > 5x - 5
Now you want to get all the x terms on the left side of the equation and all the constants on the right side of the equation.
Subtract 20 from both sides of the equation and subtract 5x from both sides of the equation to get:
6x - 5x > -5 - 20
Simplify to get
x > -25
That's your answer.
Notice that, since you did not have to multiply (or divide) both sides of the equation by a negative number, the inequality did not reverse.
To confirm this answer is good, substitute it in your original equation to see if that equation is true.
Your original equation is:
(1/6)*(3x+10 > (5/12)*(x-1)
If x = -25, this equation becomes:
(1/6)*(-65) > (5/12)*(-26)
Simplify to get:
-65/6 > -130/12 which is equivalent to:
-65/6 > -65/6
They are equal to the equation is false when x = -25.
The equation should be false when x = -25, because the answer is that x > -25.
Pick any value of x > -25 and the equation should be true.
We'll try something simple, like -20
The original equation is:
(1/6)*(3x+10 > (5/12)*(x-1)
When x = -20, this becomes:
(1/6)*(-50) > (5/12)*(-21)
Simplify to get:
-50/6 > -105/12
This is equivalent to:
-100/12 > -105/12
-100/12 is greater than -105/12 (It is more positive), so the equation is true which confirms that as long as x > -25, the original equation will be true, and as long as x is not > -25, the original equation is false.
Your answer is that x > -25.
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