Solve the inequality 4x + 8 < 20 - 2x.

Step-by-step solution

1. 1

   Decide which side collects the variable. The $x$ coefficients are $+4$ on the left and $-2$ on the right. Moving the $-2x$ to the left gives $4x+2x=6x$, a positive coefficient, which means the final division will not flip the inequality sign. Choosing the other direction ($-6x>-12$) works too but forces a flip.

2. 2

   Add $2x$ to both sides. Whatever you do to one side you do to the other:

   $4x+2x+8<20\ \Longrightarrow\ 6x+8<20$

3. 3

   Subtract 8 from both sides. Adding or subtracting a number never changes the direction of an inequality:

   $6x<12$

4. 4

   Divide by 6. The divisor is positive, so $<$ stays $<$:

   $x<2,\qquad\text{i.e. }(-\infty,2)$

   Only division or multiplication by a negative number reverses the sign — that rule is not triggered anywhere in this problem.

5. 5

   Check a point inside and the boundary. At $x=0$: $8<20$ ✓. At $x=2$: $16<16$ is false, so $2$ is correctly excluded. At $x=3$: $20<14$ is false ✓. The solution is every real number strictly less than $2$.