Question

Describe and correct the error. $$\begin{array}{l}x+4<1 \\\x+4>4<1-4 \\\x>-3\end{array}$$

Short answer

The corrected inequality should be \(x < -3\) instead of \(x > -3\).

Step-by-step solution

Step 1. Correct Inequality Sign

The major mistake is in the second line where the inequality signs don't make sense, the correct version of the line should be \((x+4) - 4 < 1 - 4\). Subtract 4 from both sides of the inequality to isolate \(x\).

Step 2. Solve for x

Once you have the correct inequality, you can simplify it to get \(x < -3\).

Step 3. Check Solution

To ensure that \(x < -3\) is the correct solution to the initial inequality \(x + 4 < 1\), substitute \(x\) with any real number less than -3 in the original inequality and it should hold true.

Step 4. Conclusion

The original deduction made by the student \(x > -3\) is thus incorrect. The correct solution is \(x < -3 \).

Key concepts

Inequality Correction

When working with inequalities, a common mistake is to mishandle the direction of the inequality sign after performing operations on both sides. Proper inequality correction involves carefully tracking how each step of manipulation affects the relationship between the quantities in the inequality. For instance, when solving the inequality \(x+4<1\), subtracting 4 from both sides yields \(x<1-4\), which simplifies to \(x<-3\).

A notable operation that can trip students up is multiplying or dividing by a negative number, which reverses the inequality sign. Always remember that adding or subtracting the same number from both sides does not change the direction of the inequality, only the values change.

Algebraic Problem-Solving

Effective algebraic problem-solving is not just about knowing the rules but understanding why they make sense. For students tackling algebraic inequalities, conceptual clarity is key. For example, consider the rule 'whatever you do to one side of the equation, do to the other.' This is grounded in the concept of balance.

Visualize a balanced scale where the two sides of an equation are equal weights. To maintain balance (equality), you must add or subtract the same weight from both sides. In the context of inequalities, we are dealing with 'less than' or 'more than' instead of balance, but the same principle of performing equal operations on both sides still applies.

Isolating Variables

The process of isolating variables is a foundational technique in solving both equations and inequalities. It involves using algebraic operations to get the variable by itself on one side of the equation or inequality sign. To do this, we often perform inverse operations, such as adding to counteract subtraction or dividing to counteract multiplication.

In our inequality example, the goal is to isolate \(x\). This is achieved by subtracting 4 from both sides of \(x+4<1\) to get \(x<1-4\), thus effectively isolating \(x\) to one side. It's critical to perform these steps methodically to ensure accuracy and to understand the underlying logic, which will help solve more complex problems in the future.