To solve the problem, we need to find an angle such that four times its complement is [tex]\(10^{\circ}\)[/tex] less than twice its complement.
First, let's recall that the complement of an angle [tex]\(x\)[/tex] is given by [tex]\(90^{\circ} - x\)[/tex].
Let's denote the unknown angle as [tex]\(x\)[/tex] degrees.
According to the problem, four times the complement of the angle ([tex]\(90^{\circ} - x\)[/tex]) is [tex]\(10^{\circ}\)[/tex] less than twice the complement of the angle ([tex]\(90^{\circ} - x\)[/tex]).
Formulating this condition into an equation:
[tex]\[ 4 \times (\text{complement}) = 2 \times (\text{complement}) - 10^{\circ} \][/tex]
Substitute [tex]\((90^{\circ} - x)\)[/tex] for the complement:
[tex]\[ 4(90 - x) = 2(90 - x) - 10 \][/tex]
Now, let's perform the algebraic steps to solve for [tex]\(x\)[/tex]:
1. Distribute the constants through the parentheses:
[tex]\[ 360 - 4x = 180 - 2x - 10 \][/tex]
2. Simplify the equation:
[tex]\[ 360 - 4x = 170 - 2x \][/tex]
3. Collect all [tex]\(x\)[/tex] terms on one side and the constants on the other side:
[tex]\[ 360 - 170 = 4x - 2x \][/tex]
[tex]\[ 190 = 2x \][/tex]
4. Solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{190}{2} \][/tex]
[tex]\[ x = 95 \][/tex]
Thus, we get [tex]\(x = 95^{\circ}\)[/tex].
So, here are the steps:
1. [tex]\(x\)[/tex] is the angle.
2. The complement of [tex]\(x\)[/tex] is [tex]\(90^{\circ} - x\)[/tex].
3. Formulated equation: [tex]\(4 \times (90 - x) = 2 \times (90 - x) - 10\)[/tex]
4. Solved for [tex]\(x\)[/tex] and found [tex]\(x = 95^{\circ}\)[/tex].
Hence, the solution does not match any options given, implying a possible discrepancy in the provided choices or an error in problem interpretation.
