Absolute value equations (ACT/SAT). Game on.

Factor: $|3x - 9| = |3(x - 3)| = |3|\,|x - 3| = 3|x - 3|$. Since $3|x - 3| = 24$, divide by $3$ to get $|x - 3| = 8$. The values of $x - 3$ are $8$ and $-8$; the negative value is $-8\textsf{.}$

Rewrite $|5x + 10| = |5(x + 2)| = |5|\,|x + 2| = 5|x + 2|$. Since $5|x + 2| = 35$, divide by $5$ to get $|x + 2| = 7$. The expression $x + 2$ equals $7$ or $-7$; the positive value is $7\textsf{.}$

Either $x - 7 = 15$, giving $x = 22$, or $x - 7 = -15$, giving $x = -8$. The positive value of $x$ is $22\textsf{.}$

Rewrite $|2x - 8| = |2(x - 4)| = |2|\,|x - 4| = 2|x - 4|$. Since $2|x - 4| = 18$, divide by $2$ to get $|x - 4| = 9$. The expression $x - 4$ equals $9$ or $-9$; the negative value is $-9\textsf{.}$

Rewrite $|6x + 12| = |6(x + 2)| = |6|\,|x + 2| = 6|x + 2|$. Since $6|x + 2| = 48$, divide by $6$ to get $|x + 2| = 8$. The expression $x + 2$ equals $8$ or $-8$; the positive value is $8\textsf{.}$

Rewrite $|2x + 10| = |2(x + 5)| = |2|\,|x + 5| = 2|x + 5|$. Since $2|x + 5| = 26$, divide by $2$ to get $|x + 5| = 13$. The expression $x + 5$ equals $13$ or $-13$; the positive value is $13\textsf{.}$

Rewrite $|4x + 16| = |4(x + 4)| = |4|\,|x + 4| = 4|x + 4|$. Since $4|x + 4| = 52$, divide by $4$ to get $|x + 4| = 13$. The expression $x + 4$ equals $13$ or $-13$; the negative value is $-13\textsf{.}$

Either $x - 2 = 11$, giving $x = 13$ and $2x = 26$, or $x - 2 = -11$, giving $x = -9$ and $2x = -18$. The positive value of $2x$ is $26\textsf{.}$