- The inequality at step 3 is \(-6x - 8 < -2\).
- To isolate the term \(-6x\), we add \(8\) to both sides of the inequality: \(-6x - 8 + 8 < -2 + 8\).
- This results in \(-6x < 6\), which is the inequality at step 4.
- Adding the same value to both sides of an inequality is justified by the addition property of inequality.
- The correct answer is \(\boxed{\text{addition property of inequality}}\).
Explanation
- Analyzing the Transformation from Step 3 to Step 4
The problem asks to identify the property used to move from step 3 to step 4 in the solving process of the inequality \(-6(x+3)+10 < -2\).
In step 3, we have the inequality:
\[-6x - 8 < -2\]
In step 4, we have the inequality:
\[-6x < 6\]
To reach step 4 from step 3, we observe the change on both sides of the inequality. The term \(-8\) on the left-hand side of step 3 is eliminated to get \(-6x\) in step 4. This is achieved by adding \(8\) to both sides of the inequality:
\[(-6x - 8) + 8 < (-2) + 8\]
\[-6x + 0 < 6\]
\[-6x < 6\]
The property that allows adding the same value to both sides of an inequality without changing the solution set is the addition property of inequality.
- Conclusion
The transition from step 3 to step 4 is completed by adding 8 to both sides. The addition property of inequality states that if \(a < b\), then \(a + c < b + c\) for any real number \(c\). By applying this property with \(c = 8\), the transformation is justified. Therefore, the property that justifies the work shown between step 3 and step 4 is the addition property of inequality.
Examples
In real-world budgeting, if you have a debt of \(\$8\) and a current balance of \(\$-2\), and you receive a credit of \(\$8\), your new balance becomes \(\$6\). This mirrors the mathematical process of adding the same quantity to both sides of an inequality to isolate a variable term, helping you track financial adjustments efficiently and correctly. Maintaining balance in equations, just like in your bank account, ensures your calculations remain accurate and fair.