Solve the inequality -4x + 6 <= 18.

Step-by-step solution

1. 1

   Move the constant first. Subtract $6$ from both sides. Adding or subtracting never affects the direction of an inequality:

   $-4x\le 12$

2. 2

   Divide by $-4$ and reverse the sign. Dividing by a negative number flips $\le$ into $\ge$:

   $x\ge\frac{12}{-4}=-3$

   Why the flip happens: multiplying by a negative reflects the number line, so the order of any two numbers is reversed. For instance $2\le 3$ but $-2\ge -3$.

3. 3

   Alternative route that avoids the flip. You can instead add $4x$ to both sides and subtract $18$:

   $6-18\le 4x\;\Longrightarrow\;-12\le 4x\;\Longrightarrow\;-3\le x$

   which is the same statement written the other way round. Keeping the variable coefficient positive is a reliable way to sidestep sign errors.

4. 4

   Check a value on each side of $-3$. At $x=0$: $-4(0)+6=6\le 18$ $\checkmark$ (should be in the solution set). At $x=-10$: $40+6=46\not\le 18$ $\checkmark$ (should be out). At the boundary $x=-3$: $12+6=18\le 18$ $\checkmark$, so the endpoint is included.