Question

Solve the inequality $\frac{2}{5}z-\frac{1}{3}z<\frac{1}{15}z-\frac{3}{5}$, graph the solution on the number line, and write the solution in interval notation.

Short answer

There is no solution of the given inequality$\frac{2}{5}z-\frac{1}{3}z<\frac{1}{15}z-\frac{3}{5}$.

Step-by-step solution

Step 1. Given information

We have given the following inequality :-

$\frac{2}{5}z-\frac{1}{3}z<\frac{1}{15}z-\frac{3}{5}$.

We have to solve the inequality for values of $x$.

We will be apply some arithmetic operations, that will convert the quality to a inequality from which we easily determine the interval of solution.

z-z<z-z-9⇒0<-9Step 2. Solve the inequality

Multiply both sides with LCD 15, to clear the fractions, then we have :-

$$\begin{gathered} 15(\frac{2}{5}z-\frac{1}{3}z)<15(\frac{1}{15}z-\frac{3}{5}) \\ \Rightarrow 6z-5z<z-9 \\ \Rightarrow z<z-9 \end{gathered}$$

Now subtract $z$from both sides, then we have :-

$$\begin{gathered} z-z<z-z-9 \\ \Rightarrow 0<-9 \end{gathered}$$

This is a contraction.

Step 3. Conclusion

As there is a contraction while solving the inequality.

So that the given inequality is not true.

Hence we can conclude that the given inequality hasno solution.

We can draw the solution graph as following :-