Which inequality shows all possible solution for X?
in pic shown its a right triangle leg side A is unknown and base is 8x+18 and hypotenuse is 12x-6
Answer choices A: x>6
B: x>_ 6
C: 1/2<x<6
D: 1/2<x<_6
Please help and explain step by step. Thanks in advance.
hypotenuse > base
so
12 x - 6 > 8 x + 18
4 x > 24
x > 6
Thank you so much!
To find the inequality that shows all possible solutions for X in the given right triangle problem, we need to consider the conditions that need to be satisfied.
In a right triangle, the Pythagorean theorem states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
Here, we have the following information about the right triangle:
- The length of one leg (side A) is unknown.
- The length of the base is 8x + 18.
- The length of the hypotenuse is 12x - 6.
We can set up the Pythagorean theorem equation as follows:
(length of hypotenuse)^2 = (length of leg A)^2 + (length of base)^2
(12x - 6)^2 = A^2 + (8x + 18)^2
Expanding and simplifying this equation, we have:
144x^2 - 144x + 36 = A^2 + 64x^2 + 288x + 324
Combining like terms, we get:
144x^2 - 144x + 36 = A^2 + 64x^2 + 288x + 324
80x^2 + 432x + 288 = A^2
To find the possible values of x, we can solve this quadratic inequality:
80x^2 + 432x + 288 ≥ 0
Now, let's solve this inequality step by step:
Step 1: Rewrite the inequality in the form of a quadratic equation:
80x^2 + 432x + 288 = 0
Step 2: Factorize the quadratic equation if possible. In this case, we cannot easily factorize it.
Step 3: Use the quadratic formula to find the values of x:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, we have a = 80, b = 432, and c = 288. Plugging these values into the formula, we can find the values of x.
Step 4: Calculate the discriminant (b^2 - 4ac) to determine the nature of the solutions.
If the discriminant is greater than zero, there are two real solutions.
If the discriminant is equal to zero, there is one real solution.
If the discriminant is less than zero, there are no real solutions.
Step 5: Analyze the solutions and determine the inequality based on the nature of the solutions.
By calculating the discriminant, we find that it is greater than zero. This means that there are two real solutions for x.
Therefore, the correct answer choice is D: 1/2 < x < 6, which signifies that x should lie between 1/2 and 6 for the given right triangle.