find the range of possible values for x.\nthe range is \\(\\square < x < \\square\\).\n(simplify your…

Explanation:

Step1: Analyze triangle sides and angles

In the two triangles, two sides are equal (marked with ticks), and we have sides 6 and 7. The angle opposite the longer side (7) is (35^\circ), and the angle opposite 6 is ((x - 8)^\circ). By the Hinge Theorem (if two sides of one triangle are congruent to two sides of another triangle, but the third side is longer, then the included angle is larger), we know that if the side opposite ((x - 8)^\circ) (which is 6) is shorter than the side opposite (35^\circ) (which is 7), then ((x - 8)^\circ< 35^\circ). Also, angles in a triangle must be positive, so ((x - 8)^\circ>0^\circ).

Step2: Solve the lower bound inequality

For the lower bound, since the angle must be positive: (x - 8>0) (x>8)

Step3: Solve the upper bound inequality

For the upper bound, using the Hinge Theorem (since 6 < 7, the angle opposite 6 is less than the angle opposite 7): (x - 8<35) (x<35 + 8) (x<43)

Answer:

(8 < x < 43)