Question

Solve the inequality for \(x\). $$ e^{1-x}<6 $$

Short answer

The values of \(x\) that satisfy the inequality are all \(x > 1 - \ln{6}\).

Step-by-step solution

Apply Natural Logarithm to Both Sides

We start off by simplifying the problem. In order to remove the \(e\) (the base of the natural logarithm) in the term on the left, we apply the natural logarithm (ln) on both sides of the inequality. By doing this, we have \(\ln{(e^{1-x})} < \ln{6}\).

Simplify Left Side

Now we simplify the left side. The logarithm rule \(\ln(a^b) = b \ln(a)\) allows us to bring down the \(1-x\) in front. So we have \((1-x) \cdot \ln{(e)} < \ln{6}\). But because \(\ln{(e)} = 1\), it is simplified to \(1-x < \ln{6}\).

Solve for \(x\)

Finally, we can solve the inequality for \(x\). By subtracting 1 from both sides and then multiplying by -1, we flip the inequality sign and solve for \(x\). Thus, our solution is \(x > 1 - \ln{6}\).