A triangle has side lengths of x, 2x, and 15, where x is an integer. What is the total num… — ACT

A triangle has side lengths of $x$ , $2x$ , and $15$ , where $x$ is an integer. What is the total number of possible values for $x$ ?

A
7
B
8
9Answer
D
10
E
11

Answer

Applying the Triangle Inequality Theorem yields three inequalities:

x + 2x > 15

(which simplifies to

x > 5

x + 15 > 2x

(which simplifies to

x < 15

), and

2x + 15 > x

(which is always true since

x

is positive). Combining these results in

5 < x < 15

. The integers in this range are

\{6, 7, 8, 9, 10, 11, 12, 13, 14\}

. Counting them gives 9 possible values.

Step-by-Step Solution

Set up the three inequalities using the Triangle Inequality Theorem for the sides

x

2x

, and

15

The inequalities are

x + 2x > 15

x + 15 > 2x

, and

2x + 15 > x

The sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side.

Solve each inequality for

x

3x > 15 \implies x > 5

, and

15 > x \implies x < 15

. The third inequality

x > -15

is always true for positive side lengths.

To find the range of valid values for

x

Combine the inequalities and count the possible integer values for

x

The combined range is

5 < x < 15

. The integer values are

\{6, 7, 8, 9, 10, 11, 12, 13, 14\}

, which gives

14 - 6 + 1 = 9

values.

Since

x

is specified as an integer, we must identify and count all integers strictly between 5 and 15.

Key Concept

Triangle Inequality Theorem

Alternative Method

Instead of algebraic manipulation, we can test values of

x

directly. For

x=5

, the sides are

5, 10, 15

, but

5+10=15

, which does not form a triangle. For

x=15

, the sides are

15, 30, 15

, but

15+15=30

, which also does not form a triangle. Testing integer values between

5

and

15

confirms they all satisfy the triangle inequality, resulting in 9 valid integers.

Estimated Time:1m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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