Find the smallest value of $x$ that satisfies the equation $|3x+7|=26$.
I notice that the equation involves the absolute value of an expression, which means that the expression inside the absolute value bars can be either positive or negative. So I can rewrite the equation as two separate cases: either $3x+7=26$ or $3x+7=-26$. I can solve for $x$ in each case by subtracting $7$ from both sides and dividing by $3$. This gives me $x=\frac{19}{3}$ or $x=-11$. To find the smallest value of $x$, I compare these two solutions and see that $-11$ is smaller than $\frac{19}{3}$. So the smallest value of $x$ that satisfies the equation is $-11$. 