Chapter 2.3, Problem 36E

To determine

To explain:

The solving of inequality equation $\frac{4}{x}\ge 2$ leads to an error by multiplying by $x$

Answer to Problem 36E

Multiply $x$ on both sides of $\frac{4}{x}\ge 2$ gives another inequality equation $4\ge 2x$ instead of solution set, so the solving of inequality equation $\frac{4}{x}\ge 2$ leads to an error when it multiplies by $x$ .

Explanation of Solution

Given information:

  $\frac{4}{x}\ge 2$

The inequality equation is given as:

  $\frac{4}{x}\ge 2$  .. . . . (1)

Now, as per given problem statement need to multiply by $x$ on both sides of first equations:

  $\begin{array}{l}\frac{4}{x}\ge 2\\ \frac{4}{x}\times x\ge 2\times x\\ 4\ge 2x\end{array}$

So, it does solve for the solution value for $x$ . Multiply $x$ on both sides of $\frac{4}{x}\ge 2$ gives another inequality equation instead of solution set.

That’s why multiplication $x$ on both sides of given inequality leads to error. The correct method for solving is:

  $\frac{4}{x}\ge 2$

Multiply by $\frac{1}{4}$ on both sides of equation:

  $\begin{array}{l}\frac{4}{x}\ge 2\\ \frac{4}{x}\times \frac{1}{4}\ge 2\times \frac{1}{4}\\ x\ge \frac{1}{2}\end{array}$

Therefore, the solution of $\frac{4}{x}\ge 2$ is $x\ge \frac{1}{2}$ .