Write an inequality to describe the possible values of x.
Help please thank you?
Easy, just use the Pythagorean theorem. It states that for a right triangle, two legs both squared and summed equals the hypotenuse squared. A wider angle makes way to a longer hypotenuse, so our theorem is adjusted:
$\displaystyle a^2+b^2 \le c^2$
We then have...
$\displaystyle (5x+3)^2 + 25x^2 \le c^2$
and...
$\displaystyle (3x+17)^2 + 25x^2 \le c^2$
BTW, I don't understand from the drawing how two sides are equal (5x)? Nevertheless, I will continue with the algebra.
$\displaystyle (5x+3)^2 + 25x^2 = (3x+17)^2 + 25x^2$
$\displaystyle 5x+3=3x+17$
$\displaystyle 2x=14$
$\displaystyle x=14$
note that for any triangle, the sum of the two sides is always greater than the third side and the absolute difference of the two sides is less than the third.. also, the measure of angle is proportional to the length of the sides, that is, if angle A is greater than angle B, then the side opposite angle A must be longer than the side opposite angle B..
note, that there was a $\displaystyle 90^o$ and was divided into $\displaystyle 60^o$ and $\displaystyle 30^o$.. so, 30 + 95 = 125 subtracted from 180 gives 55.. thus, $\displaystyle 3x + 17 > 5x$, which implies $\displaystyle 2x < 17$ or $\displaystyle x < 8.5$
for the lower bound, we have x cannot be non-positive, otherwise, we would have a negative o zero length given by 5x.. thus, we can assume that $\displaystyle 0 < x < 8.5$