What is the value of x? Justify each step. AC = 32 2x 6x + 8 A B C Drawing not to scale. AB+BC = AC a. 2x+6x+8=32 b. 8x+8=32 c. 8x = 24 d. x = 3 e. (5 points)
The Correct Answer and Explanation is:
To find the value of xx, we use the fact that the total length from point AA to point CC is the sum of the lengths of segments ABAB and BCBC. Given:
- AB=2xAB = 2x
- BC=6x+8BC = 6x + 8
- AC=32AC = 32
We can now write and solve the equation step-by-step, justifying each move.
Step-by-Step Solution and Justification
a. Write the equation from the segment addition postulate: AB+BC=AC⇒2x+6x+8=32AB + BC = AC \Rightarrow 2x + 6x + 8 = 32
This step uses the Segment Addition Postulate, which states that the sum of the parts of a segment equals the whole.
b. Combine like terms: 8x+8=328x + 8 = 32
We combined the terms 2x+6x2x + 6x to get 8x8x.
c. Subtract 8 from both sides: 8x=248x = 24
This is done using the Subtraction Property of Equality to isolate the variable term.
d. Divide both sides by 8: x=3x = 3
This uses the Division Property of Equality to solve for xx.
Final Answer:
e. x=3\boxed{x = 3}
Explanation
This problem tests your understanding of segment addition and solving linear equations. You are given the total length of a segment ACAC and expressions for the lengths of the two smaller segments ABAB and BCBC. The Segment Addition Postulate tells us that if point BB lies between points AA and CC, then AB+BC=ACAB + BC = AC. Using this principle, we form an equation that represents the total distance in terms of xx.
Substituting the given expressions into the equation, we get 2x+6x+8=322x + 6x + 8 = 32. This equation is simplified by combining like terms, yielding 8x+8=328x + 8 = 32. Next, we isolate the variable by subtracting 8 from both sides, resulting in 8x=248x = 24. Finally, we divide by 8 to solve for xx, giving us x=3x = 3.
Each step follows algebraic rules that maintain the balance of the equation. These steps not only help find the correct answer but also demonstrate a logical and methodical approach to problem-solving. This type of problem is common in geometry and algebra, emphasizing how mathematical reasoning and properties work together to find unknown values. It also reflects the importance of justifying every step, especially in more complex math, where each move builds on the previous one.
