Solve the inequality (1/3)x < -4/3.

Step-by-step solution

1. 1

   Identify the single obstacle. The variable already sits alone on the left; the only thing between us and $x$ is the coefficient $\frac13$. Undoing it means multiplying by its reciprocal, $3$.

2. 2

   Check the sign of the multiplier before using it. For inequalities the rule is: multiplying or dividing by a positive number preserves the direction of $<$, while a negative number reverses it. Here the multiplier is $3>0$, so the $<$ stays a $<$. (Had we solved $-\frac13x<-\frac43$ instead, multiplying by $-3$ would have turned it into $x>4$.)

3. 3

   Multiply both sides by 3.

   $3\cdot\frac13 x<3\cdot\left(-\frac43\right)\;\Longrightarrow\;x<-4$

   The $3$ cancels the $\frac13$ on the left, and on the right $3\cdot\frac43=4$, carrying the minus sign along.

4. 4

   Test one point inside and one outside. Take $x=-5$: the left side is $-\frac53\approx -1.667$, and $-1.667<-1.333$ is true, so $-5$ is a solution as predicted. Take $x=-3$: $-1<-1.333$ is false, correctly excluded. At the boundary $x=-4$ both sides equal $-\frac43$, and the inequality is strict, so $-4$ itself is not included.

5. 5

   State the answer as an interval. The solution set is

   $x<-4,\qquad\text{i.e.}\qquad x\in(-\infty,-4)$