Solve the inequality (x - 3)/4 < 6 - (3 - 4x)/2 and give the solution set.

Step-by-step solution

1. 1

   Multiply by the least common denominator, not by each fraction separately. The denominators are $4$ and $2$, so $\operatorname{lcm}(4,2)=4$. Multiplying an inequality by the positive number $4$ leaves the direction of $<$ unchanged — that is the only reason this first move is safe. (Multiplying by a negative number would require flipping the sign.)

2. 2

   Distribute the 4 over every term on the right, including the $6$.

   $4\cdot\frac{x-3}{4}<4\cdot 6-4\cdot\frac{3-4x}{2}\ \Longrightarrow\ x-3<24-2(3-4x)$

   The commonest error here is multiplying only the fractions and leaving the $6$ alone.

3. 3

   Expand the bracket, watching the minus sign in front of it. The $-2$ multiplies both terms inside:

   $-2(3-4x)=-6+8x$

   so

   $x-3<24-6+8x\ \Longrightarrow\ x-3<18+8x$

   Note $-2\cdot(-4x)=+8x$: two negatives give a positive, which is why the $x$ term ends up on the larger side.

4. 4

   Collect the $x$ terms on the side that keeps the coefficient positive. Subtracting $x$ from both sides:

   $-3<18+7x$

   Choosing this direction avoids ever dividing by a negative number, so no sign flip will be needed.

5. 5

   Isolate $x$. Subtract $18$, then divide by the positive $7$:

   $-21<7x\ \Longrightarrow\ -3<x\ \Longrightarrow\ x>-3$

   In interval notation the solution set is $(-3,\infty)$.

6. 6

   Test one value inside and one outside. At $x=0$ (inside): left $=-\tfrac34$, right $=6-\tfrac32=4.5$, and $-0.75<4.5$ ✓. At $x=-4$ (outside): left $=-\tfrac74=-1.75$, right $=6-\tfrac{19}{2}=-3.5$, and $-1.75<-3.5$ is false ✓. The boundary $x=-3$ gives equality on both sides, confirming it is excluded.