Question

example 3
find the range of possible values for x.
5.
there is a geometric figure (a triangle - like figure with angles 120° and 115°, sides 4x - 10, 24, 40, 24) to solve for the range of x.

Explanation:

Step1: Apply triangle inequality to first triangle

Sum of two sides > third side: $40 + 24 > 4x - 10$
$64 > 4x - 10$
$74 > 4x$
$x < 18.5$

Step2: Apply triangle inequality again

$40 + (4x - 10) > 24$
$30 + 4x > 24$
$4x > -6$
$x > -1.5$

Step3: Apply triangle inequality third time

$24 + (4x - 10) > 40$
$14 + 4x > 40$
$4x > 26$
$x > 6.5$

Step4: Combine valid inequalities

$6.5 < x < 18.5$ (or $\frac{13}{2} < x < \frac{37}{2}$)

Answer:

$6.5 < x < 18.5$