# Find the Bounds of the Zeros f(x)=2/3x-12 f(x)=23x-12
Create a list of the coefficients of the function except the leading coefficient of 1.
23,1,-12
There will be two bound options, b1 and b2, the smaller of which is the answer. To calculate the first bound option, find the absolute value of the largest coefficient from the list of coefficients. Then add 1.
Arrange the terms in ascending order.
b1=|23|,|1|,|-12|
The maximum value is the largest value in the arranged data set.
b1=|-12|
The absolute value is the distance between a number and zero. The distance between -12 and 0 is 12.
b1=12+1
Add 12 and 1.
b1=13
b1=13
To calculate the second bound option, sum the absolute values of the coefficients from the list of coefficients. If the sum is greater than 1, use that number. If not, use 1.
Simplify each term.
23 is approximately 0.6‾ which is positive so remove the absolute value
b2=23+|1|+|-12|
The absolute value is the distance between a number and zero. The distance between 0 and 1 is 1.
b2=23+1+|-12|
The absolute value is the distance between a number and zero. The distance between -12 and 0 is 12.
b2=23+1+12
b2=23+1+12
Find the common denominator.
Write 1 as a fraction with denominator 1.
b2=23+11+12
Multiply 11 by 33.
b2=23+11⋅33+12
Multiply 11 and 33.
b2=23+33+12
Write 12 as a fraction with denominator 1.
b2=23+33+121
Multiply 121 by 33.
b2=23+33+121⋅33
Multiply 121 and 33.
b2=23+33+12⋅33
b2=23+33+12⋅33
Combine fractions.
Combine fractions with similar denominators.
b2=2+3+12⋅33
Multiply 12 by 3.
b2=2+3+363
b2=2+3+363
Simplify the numerator.
Add 2 and 3.
b2=5+363
Add 5 and 36.
b2=413
b2=413
Arrange the terms in ascending order.
b2=1,413
The maximum value is the largest value in the arranged data set.
b2=413
b2=413
Take the smaller bound option between b1=13 and b2=413.
Smaller Bound: 13
Every real root on f(x)=23x-12 lies between -13 and 13.
-13 and 13
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