To find the value of [tex]\( x \)[/tex] for which [tex]\(\angle A\)[/tex] and [tex]\(\angle B\)[/tex] are supplementary, given that [tex]\(\angle A = 4x - 8\)[/tex] and [tex]\(\angle B = 2x + 2\)[/tex], we need to follow these steps:
1. Recall the definition of supplementary angles. Supplementary angles are two angles whose sum is 180 degrees. Therefore, we can write the following equation:
[tex]\[
\angle A + \angle B = 180
\][/tex]
2. Substitute the expressions for [tex]\(\angle A\)[/tex] and [tex]\(\angle B\)[/tex] into the equation:
[tex]\[
(4x - 8) + (2x + 2) = 180
\][/tex]
3. Combine like terms on the left-hand side of the equation:
[tex]\[
4x - 8 + 2x + 2 = 180
\][/tex]
[tex]\[
6x - 6 = 180
\][/tex]
4. Isolate the term involving [tex]\( x \)[/tex] by adding 6 to both sides of the equation:
[tex]\[
6x - 6 + 6 = 180 + 6
\][/tex]
[tex]\[
6x = 186
\][/tex]
5. Solve for [tex]\( x \)[/tex] by dividing both sides of the equation by 6:
[tex]\[
x = \frac{186}{6}
\][/tex]
[tex]\[
x = 31
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] is [tex]\( \boxed{31} \)[/tex].
To verify, we can check the values of [tex]\(\angle A\)[/tex] and [tex]\(\angle B\)[/tex] using [tex]\( x = 31 \)[/tex]:
[tex]\[
\angle A = 4(31) - 8 = 124 - 8 = 116
\][/tex]
[tex]\[
\angle B = 2(31) + 2 = 62 + 2 = 64
\][/tex]
By adding them together:
[tex]\[
\angle A + \angle B = 116 + 64 = 180
\][/tex]
Since the sum is 180 degrees, the calculations confirm that the value of [tex]\( x = 31 \)[/tex] is correct.
