Solve the inequality -6x + 3 > -39.

Step-by-step solution

1. 1

   Isolate the variable term. Subtract $3$ from both sides. Adding or subtracting never affects an inequality's direction, so this step is safe:

   $-6x>-42$

2. 2

   Pause before dividing. The coefficient is $-6$ — negative. Dividing (or multiplying) an inequality by a negative number reverses the direction, because it reflects the number line. This is the one rule that separates inequalities from equations.

3. 3

   Divide by -6 and flip the sign.

   $\frac{-6x}{-6}<\frac{-42}{-6}\qquad\Longrightarrow\qquad x<7$

   Note both quotients are positive: a negative divided by a negative gives a positive.

4. 4

   Sanity-check the direction with a test point. Take $x=0$, which satisfies $x<7$: the left side is $-6(0)+3=3$, and $3>-39$ ✓. Now take $x=10$, which does not: $-60+3=-57$, and $-57>-39$ is false ✓. So the direction really is $<$, not $>$.

5. 5

   Check the boundary. At $x=7$: $-42+3=-39$, giving equality, not strict inequality — so $7$ itself is excluded, matching the strict $<$. The solution set is $(-\infty,7)$, and a scan of $4001$ exact rational points on $[-20,20]$ agrees at every one ✓.