Question

Explain why multiplying by \(x\) to solve the inequality \(\frac{4}{x}>2\) might lead to an error.

Short answer

Multiplying by 'x' can lead to errors because the value of 'x' can be positive, negative or zero. Neglecting the rules of inequalities by not switching the inequality sign when 'x' is negative could lead to incorrect solutions, as well as not considering the situation that 'x' could be zero, which would make the fraction undefined. Therefore, specific methods for solving fractional inequalities should be used to avoid these potential issues.

Step-by-step solution

Understand the issue

The inequality is \(\frac{4}{x}>2\). It might seem logical to multiply both sides by 'x' to eliminate the denominator, but that's where the error can occur. The value of 'x' isn't known, so 'x' could be positive, negative, or zero.

Consider negative x values

If 'x' is negative, when you multiply both sides by 'x', you should flip the inequality sign. So, \(\frac{4}{x}*x > 2*x\) would become \(4<-2x\), which is different from the original inequality when assuming 'x' is positive where it would result in \(4> 2*x\). This discrepancy can lead to incorrect solutions.

Consider x equals zero

If 'x' were zero, the original inequality would be undefined, as division by zero is undefined in mathematics. Thus, this further shows that simply multiplying both sides by 'x' without considering the specific conditions can lead to errors.