Question

Solve the following inequation and represent the solution set on the number line, $x-3-\frac{1}{7}\left(5x-17\right)>0,\forall x\in \mathrm{ℝ}$.

Answer 1

Step 1: Given inequation: $x-3-\frac{1}{7}\left(5x-17\right)>0$

$$\begin{gathered} x-3-\frac{5x}{7}+\frac{17}{7}>0 \\ \frac{7x-5x}{7}+\frac{17-21}{7}>0 \\ \frac{2x}{7}-\frac{4}{7}>0 \end{gathered}$$

Step 2: Add $\frac{4}{7}$ to both sides of the inequation.

$$\begin{gathered} \frac{2x}{7}-\frac{4}{7}+\frac{4}{7}>\frac{4}{7} \\ \frac{2x}{7}>\frac{4}{7} \end{gathered}$$

Step 3: Multiply $\frac{7}{2}$ to both sides of the inequation.

$$\begin{gathered} \frac{2x}{7}\times \frac{7}{2}>\frac{4}{7}\times \frac{7}{2} \\ x>2 \end{gathered}$$

Therefore, the solution set to the given inequation can be given by, $\left(2,\infty \right)$.

Step 4: Represent the solution set on the number line

The solution set of the given inequality can be plotted in the number line as,