Solve the inequality 4(-x - 10) < -6(2 - x) - 6x.

Step-by-step solution

1. 1

   Expand the left side.

   $4(-x-10)=-4x-40$

   Both terms inside the bracket are negative, so both products are negative.

2. 2

   Expand the right side and watch the cancellation.

   $-6(2-x)-6x=-12+6x-6x=-12$

   The variable disappears entirely from the right-hand side. That is the structural feature of this problem: what looks like a two-sided inequality in $x$ is really $-4x-40<-12$.

3. 3

   Isolate the variable term. Add $40$ to both sides:

   $-4x<28$

4. 4

   Divide by $-4$ and reverse the inequality. This is the one rule that separates inequalities from equations:

   $x>\frac{28}{-4}=-7$

   so the solution set is $x>-7$, or in interval notation $(-7,\infty)$.

5. 5

   Test values on both sides of the boundary. At $x=-6$: left $=4(6-10)=-16$, right $=-6(8)+36=-12$, and $-16<-12$ is true $\checkmark$. At $x=-8$: left $=4(8-10)=-8$, right $=-6(10)+48=-12$, and $-8<-12$ is false $\checkmark$. At the boundary $x=-7$ both sides equal $-12$, so the endpoint is correctly excluded by the strict inequality.