A triangle has side lengths of 7, 12, and 2x + 1. If x is an integer, how many possible va… — ACT

A triangle has side lengths of $7$ , $12$ , and $2x + 1$ . If $x$ is an integer, how many possible values of $x$ exist?

A
$5$
$6$ Cevap
C
$7$
D
$8$
E
$13$

Cevap

To form a valid triangle, the length of any side must be strictly less than the sum of the other two sides and strictly greater than the positive difference of the other two sides. Applying this to the side lengths

7

12

, and

2x+1

gives the inequality

12 - 7 < 2x + 1 < 12 + 7

, which simplifies to

5 < 2x + 1 < 19

. Subtracting

1

from all parts gives

4 < 2x < 18

, and dividing by

2

gives

2 < x < 9

. The integers in this open interval are

3, 4, 5, 6, 7,

and

8

, which counts to 6 possible values.

Adım Adım Çözüm

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

We obtain three inequalities: (1)

7 + 12 > 2x + 1

, (2)

7 + (2x + 1) > 12

, and (3)

12 + (2x + 1) > 7

To find the valid range for the unknown side length expression

2x + 1

Solve the three inequalities for

x

From (1),

18 > 2x \implies x < 9

. From (2),

2x + 8 > 12 \implies 2x > 4 \implies x > 2

. From (3),

2x + 13 > 7 \implies 2x > -6 \implies x > -3

. Combining the most restrictive bounds gives the interval

2 < x < 9

To isolate the variable

x

and establish its upper and lower bounds.

Identify and count all integers

x

that satisfy the inequality

2 < x < 9

The integers strictly between 2 and 9 are

3, 4, 5, 6, 7

, and

8

. There are 6 such integers.

To find the number of possible integer values for

x

as requested by the question.

Anahtar Kavram

Triangle Inequality Theorem

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

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