Question 270154: Please help me solve this step by step:
5|5x-7|is greater than or equal to 110
I have to graph it and put into interval also.
Thanks
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! Your equation is:
5*|5x-7| >= 110
If the expression within the absolute value sign is positive, then the expression becomes:
5*(5x-7) >= 110
Simplify this equation to get:
25x - 35 >= 110
Add 35 to both sides of this equation to bet:
25x >= 110+35 = 145
Divide both sides of this equation by 25 to get:
x >= 145/25 = 5.8
If the expression within the absolute value sign is negative, then the equation becomes
5 * (-(5x-7)) >= 110
Simplify this equation to get:
5 * (-5x + 7) >= 110
Simplify further to get:
-25*x + 35 >= 110
Subtract 35 from both sides of this equation to get:
-25*x >= 110-35 = 75
Divide both sides of this equation by -25 to get:
x <= 75/-25 = -3
The answer to your question should be
x >= 5.8 or x <= -3
You have to confirm these answers are good by testing with values inside the limits and outside the limits.
When x = 5.8, 5 * |5x-7| >= 110 becomes 5 * |29-7| = 5*22 = 110 which is true because the equation says >= and this is =.
When x = -3, 5* |5x-7| >= 110 becomes 5 * |-15-7| = 5*|-22| = 5*22 = 110 which is also true because the equation says >= and this is =.
When x = 0, 5*|5x-7| >= 110 becomes 5*|-7| = 5*7 = 35 which is NOT true because 35 is not greater than or equal to 110.
Since when x = 0, it is NOT greater than 5.8 and it is NOT less than -3, this is good because the equation should NOT have been satisfied when x = 0.
When x = -5, 5 * |5x-7| >= 110 becomes 5 * |-25-7| becomes 5 * |-32| becomes 5 * 32 = 160 >= 110 which is true.
When x = 7, 5 * |5x-7| >= 110 becomes 5 * |35-7| becomes 5 * |28| becomes 5 * 28 = 140 >= 110 which is true.
The answer looks good.
When we are within the stated limits of x, our equation becomes true.
When we are outside the stated limits of x, our equation becomes false.
Your answer is:
x >= 5.8 or x <= -3
In terms of intervals this would appear as (-infinity,-3] union [5.8,infinity}
A graph of your equation would look like this:
If you draw a vertical line at x = -3 and a vertical line at x = 5.8, you will see that the value of y is greater than or equal to 110 to the left of the x = -3 vertical line and to the right of the x = 5.8 vertical line.
All of this stems from the basic definition of absolute value that states:
absolute value of x = x if x is positive, and equals -x if x is negative.
The "x" in this case is the expression (5x-7) that is within the absolute value sign.
the absolute value of 5x-7 equals 5x-7 if the expression 5x-7 is positive.
the absolute value of 5x-7 equals -(5x-7) if the expression 5x-7 is negative.
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