Solve the following inequality and check for extraneous solutions.
\[
\sqrt{5 x-4}< 3 x-7
\]
Choose the appropriate answer and fill in the values. Round your answer to three decimal places.
$x< $
$x> $
$< x \leq$
$\leq x< $
Answer
\(\boxed{\text{The inequality } \sqrt{5 x-4}<3 x-7 \text{ has no solution.}}\)
Step 1 ：Start by squaring both sides of the inequality to eliminate the square root. Be cautious as this can sometimes introduce extraneous solutions.
Step 2 ：Simplify the equation and solve for x.
Step 3 ：The solutions obtained are \(x = \frac{47}{18} - \frac{\sqrt{301}}{18}\) and \(x = \frac{\sqrt{301}}{18} + \frac{47}{18}\).
Step 4 ：Check for extraneous solutions by substituting the solutions back into the original inequality.
Step 5 ：Neither of the solutions are valid when substituted back into the original inequality.
Step 6 ：\(\boxed{\text{The inequality } \sqrt{5 x-4}<3 x-7 \text{ has no solution.}}\)
