Question

Solve the following inequation and represent the solution set on the number line, $1-\frac{4}{25}\left(2x-1\right)<1,\forall x\in \mathrm{\mathbb{N}}$ and $x<5$.

Answer 1

Step 1: Given inequations: $1-\frac{4}{25}\left(2x-1\right)<1,\forall x\in \mathrm{\mathbb{N}}$ and $x<5$.

Consider the inequality: $1-\frac{4}{25}\left(2x-1\right)<1$

Subtract $1$ to both sides of the inequality.

$$\begin{gathered} 1-\frac{4}{25}\left(2x-1\right)-1<1-1 \\ -\frac{8x}{25}+\frac{4}{25}<0 \end{gathered}$$

Step 2: Subtract $\frac{4}{25}$ to both sides of the inequality.

$$\begin{gathered} -\frac{8x}{25}+\frac{4}{25}-\frac{4}{25}<-\frac{4}{25} \\ -\frac{8x}{25}<-\frac{4}{25} \end{gathered}$$

Step 3: Multiply both sides of the inequality by $\left(-\frac{25}{8}\right)$ and change the sign of inequality.

$$\begin{gathered} -\frac{8x}{25}\times \left(-\frac{25}{8}\right)>-\frac{4}{25}\times \left(-\frac{25}{8}\right) \\ x>\frac{1}{2} \end{gathered}$$

Thus, the solution set for the given equation can be given by, $\left(\frac{1}{2},\infty \right)$.

Step 4: Solution of both equation

Now, consider the inequality $x<5$.

Thus, the solution set of the given inequation can be given by, $\left(-\infty ,5\right)$.

Hence, the solution set of $\left(\frac{1}{2},\infty \right)$ and $\left(-\infty ,5\right)$ is $\left(\frac{1}{2},5\right)$.

It is also given that $x\in \mathrm{\mathbb{N}}$.

Therefore, the final solution set can be given by, $\left\{1,2,3,4\right\}$.

Step 5: Represent the solution set on the number line

The solution set of the given inequation can be plotted on the number below,