A triangle has side lengths of 7, x+2, and 2x-1, where x is an integer. What is the total … — ACT

A triangle has side lengths of $7$ , $x+2$ , and $2x-1$ , where $x$ is an integer. What is the total number of possible values for $x$ ?

Answer: 7

Answer

The total number of possible integer values for

x

7

According to the Triangle Inequality Theorem, a triangle is formed if and only if the sum of any two side lengths is strictly greater than the third side length. Solving the three inequalities

7 + (x+2) > 2x-1

7 + (2x-1) > x+2

, and

(x+2) + (2x-1) > 7

gives

x < 10

x > -4

, and

x > 2

. The overlapping interval is

2 < x < 10

. The integers in this range are

\{3, 4, 5, 6, 7, 8, 9\}

, which gives a total of

7

possible integer values.

Step-by-Step Solution

Set up the three inequalities required by the Triangle Inequality Theorem.

7 + (x+2) > 2x-1

7 + (2x-1) > x+2

(x+2) + (2x-1) > 7

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

Solve the first inequality:

7 + (x+2) > 2x-1

x < 10

Simplifying the left side yields

x + 9 > 2x - 1

. Subtracting

x

and adding

1

to both sides results in

10 > x

, which means

x < 10

Solve the second inequality:

7 + (2x-1) > x+2

x > -4

Simplifying the left side yields

2x + 6 > x + 2

. Subtracting

x

and

6

from both sides results in

x > -4

Solve the third inequality:

(x+2) + (2x-1) > 7

x > 2

Simplifying the left side yields

3x + 1 > 7

. Subtracting

1

and dividing by

3

results in

x > 2

Determine the combined range for

x

and identify the valid integers.

The combined range is

2 < x < 10

. The valid integers are

3, 4, 5, 6, 7, 8,

and

9

To satisfy all three inequalities,

x

must be greater than

2

, greater than

-4

, and less than

10

, which simplifies to

2 < x < 10

Count the total number of valid integer values for

x

There are

7

integer values.

Counting the elements of the set

\{3, 4, 5, 6, 7, 8, 9\}

yields a total of

7

values.

Key Concept

Triangle Inequality Theorem

Estimated Time:1m 30s

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