Question

What is the range of possible values for x? the diagram is not scale. show all steps that you used to solve this problem in orde to receive full credit.

​

Answer 1

The diagram shows
`\( (2x)^\circ \)` and
`\( 54^\circ \)` angles. Solving for
`\( x \)` where
`\( (2x) + 54 = 180 \)`,
`\( x = 63^\circ \)`. Thus, the feasible range for
`\( x \)` is
`\(0 < x < 63^\circ\)`, matching option C :
`\(0 < x < 54\)`.

Given:

An angle is marked as
`\( (2x)^\circ \).`

Another angle in the diagram measures
`\( 54^\circ \).`

The diagram doesn't portray angles accurately in terms of scale.

Formula:

The sum of angles on a straight line is
`\( 180^\circ \)`.

Express the relationship between angles:

The marked angle
`\( (2x)^\circ \)` and
`\( 54^\circ \)` together form a straight line.

Therefore, their sum equals
`\( 180^\circ \)`:

`\[ (2x)^\circ + 54^\circ = 180^\circ \]`

Solve the equation for
`\( x \)`:

`\[ (2x)^\circ = 180^\circ - 54^\circ \]`

`\[ (2x)^\circ = 126^\circ \]`

`\[ x = (126^\circ)/(2) \]`

`\[ x = 63^\circ \]`

This calculation tells us that
`\( x = 63^\circ \)`.

Determine the possible range for
`\( x \)`:

As
`\( x \)` is half of
`\( (2x) \)`,
`\( x \)` should be less than
`\( (2x) \)`.

Given
`\( (2x)` =
`126^\circ \)` and
`\( x = 63^\circ \),`
`\( x \)` is indeed less than
`\( (2x) \)`.

Therefore, the valid range for
`\( x \)` is
`\( 0 < x < 63^\circ \)`.

Analyzing the options:

A)
`\(0 < x < 108\)` - This range exceeds the possible value of
`\( x \)`.

B)
`\(0 < x < 27\)` - This range is below
`\(63^\circ\),` so it's incorrect.

C)
`\(0 < x < 54\)` - This range is accurate as it falls within
`\(0 < x < 63^\circ\).`

D)
`\(27 < x < 180\)` - This range extends beyond the feasible value of
`\(x\)`.

Therefore, the correct range for
`\(x\)` based on the given information is
`\(0 < x < 63^\circ\)`, which aligns with option C:
`\(0 < x < 54\).`