Questions: Solve for x. AC=22, BC=x+14, and AB=x+10 Find x.

Transcript text: Solve for $x$. $AC=22, BC=x+14$, and $AB=x+10$ Find $x$.

Solution

Solution Steps

To solve for $ x $, we can use 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 remaining side. Given the sides $ AC = 22 $, $ BC = x + 14 $, and $ AB = x + 10 $, we can set up the following inequalities:

$ AC + BC > AB $
$ AC + AB > BC $
$ AB + BC > AC $

We will solve these inequalities to find the possible values of $ x $.

Step 1: Understand the Problem

We are given three line segments $ AC = 22 $, $ BC = x + 14 $, and $ AB = x + 10 $. We need to find the value of $ x $.

Step 2: Apply the Triangle Inequality Theorem

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. We will apply this theorem to the given sides.

$ AC + BC > AB $
$ AC + AB > BC $
$ AB + BC > AC $

Step 3: Set Up Inequalities

Using the given lengths, we set up the inequalities:

$ 22 + (x + 14) > x + 10 $
$ 22 + (x + 10) > x + 14 $
$ (x + 10) + (x + 14) > 22 $

Step 4: Simplify the Inequalities

Let's simplify each inequality one by one.

$ 22 + x + 14 > x + 10 $ \[ 36 + x > x + 10 \] Subtract $ x $ from both sides: \[ 36 > 10 \] This inequality is always true.
$ 22 + x + 10 > x + 14 $ \[ 32 + x > x + 14 \] Subtract $ x $ from both sides: \[ 32 > 14 \] This inequality is always true.
$ x + 10 + x + 14 > 22 $ \[ 2x + 24 > 22 \] Subtract 24 from both sides: \[ 2x > -2 \] Divide by 2: \[ x > -1 \]

Step 5: Verify the Solution

Since the only constraint we have is $ x > -1 $, and there are no other restrictions, we can conclude that $ x $ can be any value greater than $-1$.