Solve ∣7𝑥+42∣=14𝑥∣7x+42∣=14x. Identify the solution and an extraneous solution.

1. Break Down the Problem

We start with the equation $|7x + 42| = 14x$. To solve this, we need to consider two cases because of the absolute value:

1. $7x + 42 = 14x$
2. $7x + 42 = -14x$

2. Relevant Concepts

Using properties of absolute value, we can derive the two equations from $|a| = b$ which leads to $a = b$ or $a = -b$.

3. Analysis and Detail

Case 1: $7x + 42 = 14x$

Rearranging gives: $42 = 14x - 7x$ $42 = 7x$ Dividing both sides by 7: $x = 6$

Case 2: $7x + 42 = -14x$

Rearranging gives: $7x + 42 + 14x = 0$ $21x + 42 = 0$ $21x = -42$ Dividing both sides by 21: $x = -2$

4. Verify and Summarize

Now we have two potential solutions: $x = 6$ and $x = -2$.

• Check $x = 6$: $|7(6) + 42| = |42 + 42| = |84| = 84$ $14(6) = 84$ This solution is valid.

• Check $x = -2$: $|7(-2) + 42| = |-14 + 42| = |28| = 28$ $14(-2) = -28$ This solution does not hold since $28 \neq -28$. Thus, $x = -2$ is an extraneous solution.

Final Answer

The solution to the equation is $x = 6$, and $x = -2$ is an extraneous solution.