Final answer:
To find the value of x, we use the triangle inequality theorem and set up the inequality AB + BC > AC with the given side lengths. After simplifying the inequality and solving for x, we determine that x must be greater than 14.5. Thus, the smallest integer value for x is 15, which satisfies the inequality with the given side lengths. Thus, the correct answer is A. 15.
Step-by-step explanation:
The student is asked to find the value of x given the lengths of the sides of a triangle: AC = 62, BC is described by the equation BC = -1 + 4x, and AB = 5.
We can infer that this problem is likely dealing with the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Given this, we can set up the inequalities based on the sides of the triangle:
AB + BC > AC
AC + AB > BC
AC + BC > AB
However, since we only have one variable to solve for, x, and only one of these inequalities includes x, we can focus on that particular inequality.
Plugging the given values into the inequality AB + BC > AC gives:
5 + (-1 + 4x) > 62
Combining like terms and simplifying:
4 + 4x > 62
Subtract 4 from both sides of the inequality:
4x > 58
Divide both sides by 4 to solve for x:
x > 14.5
Since x must be an integer (to match the answer choices provided), and the next integer greater than 14.5 is 15, we check if 15 is the answer. Plugging 15 into the equation for BC we get:
BC = -1 + 4(15) = 59
Since 5 + 59 is indeed greater than 62, x = 15 is the correct answer to the problem.
So, the correct answer is A. 15.