Solve the inequality 1.3x - 5 > x + 3.

Step-by-step solution

1. 1

   Read the coefficient carefully. The leading coefficient is $1.3$, not $3$ - a very easy misread that changes the whole answer. Keep the decimal in place through every step.

2. 2

   Subtract x from both sides. $1.3x - x - 5 > 3 \quad\Longrightarrow\quad 0.3x - 5 > 3,$ since $1.3x - 1.0x = 0.3x$. The coefficient of $x$ is now the small positive number $0.3$.

3. 3

   Add 5 to both sides. $0.3x > 8.$

4. 4

   Divide by 0.3, keeping the direction. $0.3>0$, so the inequality sign does not flip: $x > \frac{8}{0.3} = \frac{80}{3} = 26.\overline{6}.$ Multiplying numerator and denominator by $10$ turns the decimal division into the exact fraction $\tfrac{80}{3}$.

5. 5

   Verify at a test point and at the boundary. At $x=30$: left $=1.3(30)-5 = 34$, right $=33$, and $34>33$ holds. At $x = \tfrac{80}{3}$: left $= 1.3\cdot\tfrac{80}{3}-5 = \tfrac{104}{3}-5 = \tfrac{89}{3}$ and right $= \tfrac{80}{3}+3 = \tfrac{89}{3}$ - equal, so the endpoint is correctly excluded by the strict inequality.

6. 6

   Contrast with the integer-coefficient version. If the problem were $3x-5>x+3$ the collected form would be $2x>8$ and the answer $x>4$. Because $0.3$ is roughly one-seventh of $2$, the true boundary $\tfrac{80}{3}$ is nearly seven times larger.