Question

inequalities in two triangles
find the range of values for x.
15
(2x - 10)°
48°
18
? < x <

Explanation:

Step1: Apply Hinge Theorem

Since 18 > 15, the angle opposite 18 is larger: $2x - 10 > 48$

Step2: Solve first inequality

$2x > 58 \implies x > 29$

Step3: Angle must be positive

$2x - 10 > 0 \implies 2x > 10 \implies x > 5$ (irrelevant, since x>29 is stricter)

Step4: Angle less than 180°

$2x - 10 < 180 \implies 2x < 190 \implies x < 95$? Wait, no—wait, total angle in a triangle can't have this angle exceed 180 minus other angles? No, no: the angle itself as a single angle in a triangle must be less than 180°, but actually, more accurately, in the triangle, the angle $2x-10$ must be positive and less than 180°, but also, since it's an angle in a triangle, it must be less than 180 minus the sum of other angles? Wait no, no—wait the Hinge Theorem gives the inequality between the two angles because the sides opposite are 18 and 15. So the key inequalities are:

1. $2x - 10 > 48$ (from Hinge Theorem: longer side opposite larger angle)
2. $2x - 10 < 180$ (angle can't be ≥180° in a triangle)

Wait solving 1: $2x > 58 \implies x>29$

Solving 2: $2x < 190 \implies x < 95$? Wait wait, no—wait, no, actually, in any triangle, each angle must be greater than 0 and less than 180°, but also, the sum of angles is 180°, but since we don't know the other angles, the only constraints from the angle itself are:

$0 < 2x -10 < 180$ and $2x -10 >48$

So combining: $48 < 2x -10 <180$

$48+10 <2x <180+10$

$58 <2x <190$

$29

Ah, right! So the range is 29 <x <95.

Wait I made a mistake earlier in step4—yes, the angle $2x-10$ must be greater than 48° (from Hinge) and less than 180° (since it's an angle in a triangle). So combining those gives the correct range.

So correcting:

Step1: Hinge Theorem inequality

$2x -10 >48$

Step2: Solve for lower bound

$2x >58 \implies x>29$

Step3: Angle <180°

$2x -10 <180$

Step4: Solve for upper bound

$2x <190 \implies x<95$

Final range: 29 <x <95.

Yes, that's correct. The initial mistake was a miscalculation in step4, but now fixed.

So the answer is 29 <x <95.

Answer:

29 < x < 119