Question

reasoning

1. in $\triangle abc$, side $\overline{ac}$ is three times as long as side $\overline{ab}$. side $\overline{ac}$ is the longest side of $\triangle abc$.

(a) set up an inequality based on $\overline{ac}$ being the shortest path from $a$ to $c$.
(b) if $x$ must be an integer value, then what is the largest value that $x$ could be?

Explanation:

Step1: Apply triangle inequality theorem

In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Since \(AC = 3x\), \(AB=x\), and \(BC = 10\), and \(AC\) is the shortest path from \(A\) to \(C\) (by triangle - inequality: \(AB+BC>AC\)).
So the inequality is \(x + 10>3x\).

Step2: Solve the inequality for part (a)

$$
LATEXBLOCK0
$$

Step3: Find the largest integer value of \(x\) for part (b)

Since \(x<5\) and \(x\) is an integer, the largest integer value of \(x\) that satisfies the inequality \(x + 10>3x\) is \(x = 4\). We can also check the triangle - inequality for \(x = 4\):
\(AB=x = 4\), \(BC = 10\), \(AC=3x=12\)
Check \(AB + BC>AC\): \(4 + 10=14>12\)
Check \(AB+AC>BC\): \(4+12 = 16>10\)
Check \(BC + AC>AB\): \(10 + 12=22>4\)

Answer:

(a) The inequality is \(x + 10>3x\).
(b) The largest value of \(x\) is \(4\).