the sides of a triangle have lengths x, x+4, and 20. specify those values of x for which the triangle is acute with the longest side being 20.

The Triangle Inequality Theorem states that the sum of any 2 sides of a triangle must be greater than the measure of the third side. (Think about it, if one side was the same length as the other two combined that would make a straight line. If it was any longer the other two sides couldn't reach the endpoints.)

This means:

(x) + (x + 4) > (20)

2x + 4 > 20

2x > 5

x > 2.5

It also means:

(x) + (20) > (x + 4)

and

(x + 4) + (20) > x

but the last two don't help any because they are true for all values of x, so it doesn't narrow down the answer any more.

Now you also said the triangle is acute. That means that all angles are less than 90°. Since 20 is the longest side length, the angle across from the side with length 20 is the biggest angle. So x has to be less than whatever values make that angle exactly 90. So just find the values of x when the angle opposite the side with length 20 is 90.

Pythagorean theorem says:

x² + (x+4)² = 20²

x² + (x+4)(x+4) = 400

x² + x² + 8x + 16 = 400

2x² + 8x + 16 = 400

Divide everything by 2.

x² + 4x + 8 = 200

x² + 4x - 192 = 0

Use quadratic formula to figure out that:

x = 12 and x = -16

It can't be negative because we are talking about side lengths.

So x has to be less than 12. x < 12

SO the final answer is 2.5 < x < 12

Actually I made a mistake it should be:

(x) + (x + 4) > (20)

2x + 4 > 20

2x > 16

x > 8

 

So the final answer is  8 < x < 12