Question 1199165
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Assuming x is an integer, then here are the ways to multiply to 10 using only positive integers:
1*10 = 10
2*5 = 10


The gap from x-2 to x+1 is 3 units 
Since the gap from -2 to +1 is also 3 units.
Use a number line to see this.


Because of this 3 unit gap, we ignore the factor pair 1*10.
Instead we go for 2*5 which has the 3 unit gap.


x-2 is the smaller of the x expressions.
Meaning the value of x-2 must be 2
This means x+1 must be 5


Answer: The length of the garden is 5 feet long.


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Another approach:


This is probably the more standard universal approach.


The width x-2 and length x+1 are multiplied to get the area of 10 sq ft.


width*length = area
(x-2)*(x+1) = 10
w(x+1) = 10 ...... let w = x-2
wx+w = 10 ...... distribute
x(w) + 1(w) = 10
x(x-2) + 1(x-2) = 10 .... plug in w = x-2
x^2-2x + 1x - 2 = 10
x^2-x-2 = 10
x^2-x-2-10 = 0
x^2-x-12 = 0


We can factor to solve for x
x^2-x-12 = 0
(x-4)(x+3) = 0
x-4 = 0 or x+3 = 0
x = 4 or x = -3


Or we can use the quadratic formula
{{{x = (-b+-sqrt(b^2-4ac))/(2a)}}}


{{{x = (-(-1)+-sqrt((-1)^2-4(1)(-12)))/(2(1))}}}


{{{x = (1+-sqrt(49))/(2)}}}


{{{x = (1+-  7)/(2)}}}


{{{x = (1+7)/(2)}}} or {{{x = (1-7)/(2)}}}


{{{x = (8)/(2)}}} or  {{{x = (-6)/(2)}}}


{{{x = 4}}} or  {{{x = -3}}}
Ignore the negative x value since it causes either x-2 and x+1 to be negative. 
A negative length or width makes no sense.


The only practical solution is x = 4.


If x = 4, then:
x-2 = 4-2 = 2
x+1 = 4+1 = 5
Giving us a 2 ft by 5 ft garden of area 2*5 = 10 sq ft.
The answer is confirmed.


Answer: The length of the garden is 5 feet long.
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