Question

taylor is putting a fence around his triangular flower bed. the perimeter needs to be at least 100 feet. if the sides of the triangle are ( 2x + 5 ), ( x - 4 ), and ( 2x ), which inequality could be used to find the value of ( x )?
( 2x + 5 + x - 4 + 2x > 100 )
( 2x + 5 + x - 4 + 2x geq 100 )
( 2x + 5 + x - 4 + 2x < 100 )
( 2x + 5 + x - 4 + 2x leq 100 )

Explanation:

Step1: Recall the formula for perimeter of a triangle

The perimeter \(P\) of a triangle with sides \(a\), \(b\), and \(c\) is \(P=a + b + c\). Here, \(a = 2x+5\), \(b=x - 4\), \(c = 2x\). So, \(P=(2x + 5)+(x - 4)+2x\).

Step2: Simplify the perimeter expression

Combine like - terms: \((2x+x + 2x)+(5 - 4)=5x+1\).

Step3: Analyze the inequality based on the perimeter requirement

Since the perimeter needs to be at least 100 (i.e., \(P\geq100\)), substitute \(P = 5x + 1\) into the inequality. We get \(5x+1\geq100\). The left - hand side \(5x + 1\) is equivalent to \((2x + 5)+(x - 4)+2x\). So the inequality \((2x + 5)+(x - 4)+2x\geq100\) (which is \(2x+5+x - 4+2x\geq100\)) is the correct one.

Answer:

\(2x + 5+x - 4+2x\geq100\)