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Find all $c$ such that $|c + 5| - 3|c + 7| = 10 + 2|c - 8|.$ Enter all the solutions, separated by commas.

 Dec 9, 2023
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We can expand the absolute value expressions to get:

\begin{align*} |c + 5| - 3|c + 7| &= 10 + 2|c - 8| \ \Rightarrow \qquad 10 + 2|c - 8| &= \begin{cases} c + 5 - 3(c + 7), &\text{if } c + 5 \ge 0 \ -(c + 5) - 3(c + 7), &\text{if } c + 5 < 0 \end{cases} \ \Rightarrow \qquad 10 + 2|c - 8| &= \begin{cases} -3c - 16, &\text{if } c \ge -5 \ -4c - 22, &\text{if } c < -5 \end{cases} \end{align*}

 

Therefore, we have two cases to consider:

 

Case 1: c≥−5

In this case, we have −3c−16=10+2∣c−8∣. Solving for c, we get:

\begin{align*} -3c - 16 &= 10 + 2|c - 8| \ \Rightarrow \qquad -3c &= 26 + 2|c - 8| \ \Rightarrow \qquad -3c - 2|c - 8| &= 26 \ \Rightarrow \qquad -5|c - 8| &= 26 \ \Rightarrow \qquad |c - 8| &= -\frac{26}{5} \end{align*}

 

Since the absolute value of an expression is non-negative, the equation ∣c−8∣=−526​ has no solution.

 

Case 2: c<−5

 

In this case, we have −4c−22=10+2∣c−8∣. Solving for c, we get:

 

\begin{align*} -4c - 22 &= 10 + 2|c - 8| \ \Rightarrow \qquad -4c &= 32 + 2|c - 8| \ \Rightarrow \qquad -4c - 2|c - 8| &= 32 \ \Rightarrow \qquad -6|c - 8| &= 32 \ \Rightarrow \qquad |c - 8| &= -\frac{16}{3} \end{align*}

 

Again, since the absolute value of an expression is non-negative, the equation ∣c−8∣=−316​ has no solution.

 

Therefore, there are no values of c that satisfy the equation ∣c+5∣−3∣c+7∣=10+2∣c−8∣. Thus, the answer is No solution​.

 Dec 9, 2023

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