A triangle has side lengths such that one side is 3 units less than twice an integer y, an… — ACT

A triangle has side lengths such that one side is $3$ units less than twice an integer $y$ , another side is $4$ units more than $y$ , and the third side is $11$ units. How many different triangles can be formed with these side lengths?

A
8
B
13
14Answer
D
15
E
16

Answer

The correct answer is 14. Applying the Triangle Inequality Theorem, the sum of any two sides of the triangle must be strictly greater than the third side. This gives three inequalities: (2y - 3) + (y + 4) > 11, (2y - 3) + 11 > y + 4, and (y + 4) + 11 > 2y - 3. Solving these inequalities yields y > 10/3, y > -4, and y < 18. The common intersection is 10/3 < y < 18. Since y is an integer, y can range from 4 to 17 inclusive. There are 17 - 4 + 1 = 14 integers in this range.

Step-by-Step Solution

Translate the verbal descriptions into algebraic expressions for the side lengths.

The three side lengths are represented as

2y - 3

y + 4

, and

11

To apply mathematical theorems, the side lengths must first be written in algebraic form.

Set up the three inequalities required by the Triangle Inequality Theorem, stating that the sum of any two sides must be strictly greater than the third side.

The inequalities are: 1)

(2y - 3) + (y + 4) > 11

, 2)

(2y - 3) + 11 > y + 4

, and 3)

(y + 4) + 11 > 2y - 3

The Triangle Inequality Theorem guarantees that three segment lengths can form a non-degenerate triangle.

Solve each of the three inequalities for

y

3y + 1 > 11 \implies 3y > 10 \implies y > \frac{10}{3} \approx 3.33

.
2)

2y + 8 > y + 4 \implies y > -4

.
3)

y + 15 > 2y - 3 \implies 18 > y \implies y < 18

Solving the inequalities identifies the constraints on the variable

y

Find the intersection of all three solution intervals and identify the valid integer values for

y

The intersection of the intervals is

3.33 < y < 18

. Since

y

must be an integer,

y

can be any integer from

4

17

, inclusive:

\{4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17\}

The value of

y

must simultaneously satisfy all three inequality conditions.

Count the number of integers in the inclusive range

[4, 17]

The number of integers is

17 - 4 + 1 = 14

This gives the total number of distinct triangles that can be formed.

Key Concept

Triangle Inequality Theorem

Question

Answer

Step-by-Step Solution

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