Question 421857: PLEASE solve the equation:
2*6x+5*-1=25
*= absolute value signs
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! you normally show absolute value sign as | (shift + \).
this character is shown above the enter key on my keyboard.
your problem would show up as 2 * |6x+5| - 1 = 25 if i understand you correctly.
another way to show absolute value is to use abs(6x+5).
either way works.
* is usually reserved as a multiplication symbol.
anyway, your equation is:
2 * abs(6x+5) - 1 = 25
add 1 to both sides of the equation to get:
2 * abs(6x+5) = 26
divide both sides of equation by 2 to get:
abs(6x+5) = 13
the object of all that was to isolate the absolute value part of the equation to the left side of the equation and everything else to the right side of the equation.
next, you apply the rules of absolute value.
the rule to use is:
if the expression within the absolute value sign is >= 0, then the absolute value of the expression equals the expression.
if the expression within the absolute value sign is <-, then the absolute value of the expression equals minus the expression.
the expression in this case is 6x+5.
if 6x + 5 is >= 0, then abs(6x+5) = 6x+5
if 6x + 5 is < 0, then abs(6x+5) = -(6x+5)
the equation that you want to solve for is:
abs(6x+5) = 13
when the expression is positive, the absolute value of the expression is the same as:
6x + 5 = 13
solve for x to get x = 8/6
when the expression is negative, the absolute value of the expression is the same as:
-(6x + 5) = 13
solve for x as follows:
simplify to get:
-6x - 5 = 13
add 5 to both sides of the equation to get:
-6x = 18
divide both sides of the equation by -6 to get:
x = -3
let's see how we did.
back to the original equation of:
abs(6x+5) = 13
let x = 8/6
expression becomes:
abs(6*8/6) + 5) = 13
simplify to get:
abs(8+5) = 13
simplify further to get:
abs(13) = 13
simplify further to get:
13 = 13
statement is true confirming that the equation is valid when x = 8/6.
let x = -3
expression becomes:
abs(6*(-3) + 5) = 13
simplify to get:
abs(-18 + 5) = 13
simplify further to get:
abs(-13) = 13
simplify further to get:
13 = 13
statement is true confirming that the equation is valid when x = -3
your answer is that:
the equation is true when x = 8/6, or when x = -3.
a graph of the equation y = abs(6x+5) and the equation y = 13 would look like this:
our 2 points of interest are the intersection of the graph of the equation of y = abs(6x+5) with the graph of the equation of y = 13.
if you draw a vertical line from the intersection of these 2 equations, you will see that the value of x is equal to -3 on the negative side, and equal to 8/6 on the positive side.
the graph shows all values of x and all corresponding values of y, but our focus of interest is the value of x when the value of y = 13.
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