Solve the compound inequality 5x - 4 > 3 >= 1 - x.

Step-by-step solution

1. 1

   Recognise why you cannot operate on all three parts at once. In $1<x+4\le 2$ the variable appears only in the middle, so one subtraction fixes everything. Here the variable appears in the outer parts, so the chain has to be split into two separate inequalities joined by 'and':

   $5x-4>3\qquad\text{and}\qquad 3\ge 1-x$

2. 2

   Solve the first inequality. Add $4$ then divide by the positive $5$:

   $5x>7\quad\Longrightarrow\quad x>\frac{7}{5}$

3. 3

   Solve the second inequality. Subtract $1$ from both sides to get $2\ge -x$, then multiply by $-1$ and reverse the direction:

   $-2\le x\quad\Longleftrightarrow\quad x\ge -2$

4. 4

   Intersect the two solution sets. We need $x>\tfrac75$ and $x\ge -2$ at the same time. Since $\tfrac75=1.4>-2$, the first condition is the stricter one and absorbs the second:

   $x>\frac{7}{5}$

5. 5

   Check a value inside and one that fails only the weaker part. At $x=2$: $5(2)-4=6>3$ $\checkmark$ and $3\ge 1-2=-1$ $\checkmark$. At $x=0$: $-4>3$ fails, so $0$ is out $\checkmark$. Solution set: $\left(\tfrac75,\infty\right)$.