An architect is designing a triangular window with side lengths, in feet, represented by x… — ACT

An architect is designing a triangular window with side lengths, in feet, represented by $x$ , $3x - 2$ , and $18$ . If the value of $x$ must be an integer, what is the sum of all possible values of $x$ ?

Answer: 30

Answer

The sum of all possible integer values of

x

is 30.

By applying the Triangle Inequality Theorem, the sum of any two sides of a triangle must be strictly greater than the third side. This yields the inequalities

x + (3x - 2) > 18

(which simplifies to

x > 5

x + 18 > 3x - 2

(which simplifies to

x < 10

), and

(3x - 2) + 18 > x

(which simplifies to

x > -8

). The intersection of these inequalities is

5 < x < 10

. Since

x

must be an integer, the possible values are

6

7

8

, and

9

. The sum of these values is

6 + 7 + 8 + 9 = 30

Step-by-Step Solution

Set up the first triangle inequality constraint where the sum of the two variable sides is greater than the constant side.

x + (3x - 2) > 18 \implies 4x > 20 \implies x > 5

The Triangle Inequality Theorem states that the sum of any two sides of a triangle must be strictly greater than the third side.

Set up the second triangle inequality constraint where the sum of

x

and the constant side is greater than the other variable side.

x + 18 > 3x - 2 \implies 20 > 2x \implies x < 10

To satisfy the Triangle Inequality Theorem for all combinations of sides.

Set up the third triangle inequality constraint where the sum of the second variable side and the constant side is greater than the first variable side.

(3x - 2) + 18 > x \implies 2x > -16 \implies x > -8

To ensure the third side combination is mathematically valid.

Find the intersection of all three inequalities to determine the valid range for

x

5 < x < 10

The value of

x

must satisfy all three inequalities simultaneously.

Identify the integer values of

x

within the open interval

(5, 10)

and calculate their sum.

The integers are

6, 7, 8,

and

9

. Sum =

6 + 7 + 8 + 9 = 30

The problem specifies that

x

must be an integer, so we sum only the integers strictly between 5 and 10.

Key Concept

Triangle Inequality Theorem

Estimated Time:2m 0s

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