Question

1. error analysis describe and correct the error in solving the inequality. \\(\frac{x}{3} + 4 < 6\\) \\(x + 4 < 18\\) \\(x < 14\\)

Explanation:

Step1: Identify the error

When multiplying both sides of the inequality \(\frac{x}{3}+4 < 6\) by 3 to eliminate the fraction, the error occurred in not multiplying the constant term 4 by 3. The correct step should be multiplying each term on both sides by 3.
So the incorrect step was \(x + 4<18\) (should be \(x+12 < 18\) after multiplying each term by 3).

Step2: Solve the inequality correctly

Start with the original inequality:
\(\frac{x}{3}+4 < 6\)
Subtract 4 from both sides: \(\frac{x}{3}<6 - 4\)
\(\frac{x}{3}<2\)
Multiply both sides by 3: \(x < 2\times3\)
\(x < 6\)

Or, when multiplying both sides by 3 initially:
Multiply each term by 3: \(3\times\frac{x}{3}+3\times4 < 3\times6\)
\(x + 12<18\)
Subtract 12 from both sides: \(x<18 - 12\)
\(x < 6\)

Answer:

The error was not multiplying the constant term \(4\) by \(3\) when multiplying both sides of the inequality \(\frac{x}{3}+4 < 6\) by \(3\). The correct solution is \(x < 6\).