Question

Is \(y>3 x+2 ?\) (1) \(y>4 x\) (2) \(x>2\) A. 1 alone, not 2 alone B. 2 alone, not 1 alone C. 1 and 2 together (need both) D. 1 alone or 2 alone E. 1 and 2 together are not sufficient

Short answer

A. 1 alone, not 2 alone

Step-by-step solution

- Understand the inequalities

The problem provides three inequalities: the one we need to verify, which is \(y > 3x + 2\), and two conditions, \(y > 4x\) and \(x > 2\). We need to determine which of the conditions alone or together are sufficient to verify \(y > 3x + 2\).

- Analyze condition (1): \(y > 4x\)

Rewrite the inequality: \(y > 4x\). Compare it to the inequality we need to verify \(y > 3x + 2\). Examine if \(4x > 3x + 2\). This simplifies to \(x > 2\). Therefore, condition (1) ensures \(y > 3x + 2\) when \(x > 2\).

- Analyze condition (2): \(x > 2\)

If we only have \(x > 2\), this does not guarantee anything about \(y\)'s relation to either \(3x + 2\) or \(4x\). Therefore, condition (2) alone does not ensure \(y > 3x + 2\).

- Evaluate necessity of combining conditions

Condition (1) alone was found to be sufficient. Therefore, combining condition (1) with condition (2) should also be sufficient, but it's redundant because condition (1) alone is already enough.

- Conclusion

Based on the analysis, condition (1) alone is sufficient, while condition (2) alone is not.

Key concepts

Inequalities in Algebra

Understanding inequalities is a vital part of algebra. Inequalities express a relationship where one quantity is greater than or less than another. For example, in the exercise, we deal with the inequalities: \( y > 3x + 2 \), \( y > 4x \), and \( x > 2 \). The challenge is to determine which of the given conditions might be sufficient to establish the inequality \( y > 3x + 2 \). In algebra, inequalities are often represented on a number line and can be manipulated similarly to equations, with important differences when multiplying or dividing by negative numbers. To graphically understand these inequalities, plotting them on a coordinate plane helps visualize the solutions. By shading the regions defined by each inequality, we can see common areas of intersection which represent solutions that satisfy multiple inequalities simultaneously.

Condition Analysis

Analyzing conditions involves breaking down each given condition to see if it supports the inequality we wish to confirm. In Step 2 of the solution, we examine condition (1): \( y > 4x \). We compare this to the target inequality \( y > 3x + 2 \). By testing values and rearranging the inequalities, we derive \( y > 3x + 2 \) from \( y > 4x \) when \( x > 2 \). This shows that the first condition provides sufficient information, under another condition. Meanwhile, condition (2) states \( x > 2 \). By itself, this tells us nothing about \( y \)'s relationship with \( 3x + 2 \) or \( 4x \). Thus, only the combination with another condition might help, but doesn't guarantee a solution alone. Effective condition analysis helps avoid unnecessary steps and focuses on understanding the essential relationships between the variables in the inequalities.

Sufficient Conditions

In problem-solving, identifying sufficient conditions is critical. A sufficient condition guarantees the truth of a statement. In this exercise, condition (1) is a sufficient condition for proving \( y > 3x + 2 \), because it directly leads us to this inequality when combined with another fact, \( x > 2 \). The presence of \( y > 4x \) implicitly covers the required criteria for \( y > 3x + 2 \). However, condition (2) by itself is not sufficient; while \( x > 2 \) sets a restriction on one variable, it does not provide enough information about \( y \) to verify our target inequality. This completeness means that condition (1) alone is adequate, making the combination of conditions unnecessary. Recognizing and applying sufficient conditions efficiently can streamline your problem-solving process significantly.