y gt 2x - 12x gt 5Which of the following consists of the y-coordinates of all the points that satisfy...

1. TRANSLATE the problem information

• Given system:
  • \(\mathrm{y \gt 2x - 1}\)
  • \(\mathrm{2x \gt 5}\)
• What we need: The range of y-values for all points satisfying both inequalities

2. INFER the solution strategy

• Key insight: We need to find a way to connect these inequalities
• Notice that the first inequality contains "2x - 1" and the second contains "2x"
• Strategy: Manipulate the second inequality to get it in terms of "2x - 1"

3. SIMPLIFY the second inequality

• Start with: \(\mathrm{2x \gt 5}\)
• Subtract 1 from both sides: \(\mathrm{2x - 1 \gt 4}\)
• This gives us a direct connection to the first inequality!

4. INFER the relationship using transitivity

• We now have:
  • \(\mathrm{y \gt 2x - 1}\) (first inequality)
  • \(\mathrm{2x - 1 \gt 4}\) (modified second inequality)
• By transitive property: \(\mathrm{y \gt 2x - 1 \gt 4}\)
• Therefore: \(\mathrm{y \gt 4}\)

Answer: B. \(\mathrm{y \gt 4}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the strategic connection between the inequalities. They might try to solve each inequality independently, getting \(\mathrm{x \gt 5/2}\) from the second inequality, then incorrectly think they need to substitute specific x-values into the first inequality rather than using the relationship between 2x and 2x - 1.

This may lead them to select Choice C (\(\mathrm{y \gt 5/2}\)) by incorrectly replacing x with y in their solution.

Second Most Common Error:

Poor SIMPLIFY execution: Students might make algebraic errors when manipulating the inequalities, such as incorrectly adding instead of subtracting, or making sign errors when working with the inequality relationships.

This leads to confusion and incorrect algebraic expressions, potentially causing them to select Choice A (\(\mathrm{y \gt 6}\)) or Choice D (\(\mathrm{y \gt 3/2}\)).

The Bottom Line:

This problem requires recognizing that system of inequalities problems often need creative manipulation to reveal hidden connections. The key insight is seeing how to modify one inequality to create a "bridge" to the other, then apply transitivity.