Question: 18. In the triangle, the value of $x$ is greater than 2 times the value of $y$. What are the possible values of $x$?
$x > 2y$
75°
28 Algebra 1
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Triangle shape at top-left: a slanted triangle with 756 marked at the bottom-left interior angle, $x$ marked near the top-left vertex, and $y$ marked near the bottom-right vertex.
1. **State the problem:** We have a triangle with angles $x$, $y$, and $75^\circ$. We know that $x > 2y$ and want to find the possible values of $x$.
2. **Recall the triangle angle sum rule:** The sum of interior angles in any triangle is always $$x + y + 75^\circ = 180^\circ.$$
3. **Express $x$ in terms of $y$:** From the sum rule,
$$x + y = 180^\circ - 75^\circ = 105^\circ.$$
So,
$$x = 105^\circ - y.$$
4. **Use the inequality $x > 2y$:** Substitute $x$ from above,
$$105^\circ - y > 2y.$$
5. **Solve the inequality:**
$$105^\circ > 3y$$
$$y < \frac{105^\circ}{3} = 35^\circ.$$
6. **Find possible values of $x$ using $x = 105^\circ - y$ and $y < 35^\circ$:**
Since $y$ must be less than $35^\circ$, the maximum $y$ can approach is just under $35^\circ$.
Therefore,
$$x = 105^\circ - y > 105^\circ - 35^\circ = 70^\circ.$$
7. **Conclusion:** The possible values of $x$ satisfy
$$x > 70^\circ.$$
This means $x$ must be greater than $70^\circ$ to satisfy the condition $x > 2y$ in the triangle with a $75^\circ$ angle.