|3x - 9| geq 2y + 1For which of the following tables are all the values of x and their...

1. TRANSLATE the problem requirement

• Given: The inequality \(|3\mathrm{x} - 9| \geq 2\mathrm{y} + 1\) and four tables of coordinate pairs
• Need to find: Which table has ALL coordinate pairs that satisfy the inequality
• Strategy: Test each (x, y) pair in each table by substitution

2. SIMPLIFY through systematic testing of Choice A

• For \(\mathrm{x} = 2\), \(\mathrm{y} = 0\): \(|3(2) - 9| \geq 2(0) + 1\)
  • Left side: \(|6 - 9| = |-3| = 3\)
  • Right side: \(0 + 1 = 1\)
  • Check: \(3 \geq 1\) ✓

• For \(\mathrm{x} = 3\), \(\mathrm{y} = -1\): \(|3(3) - 9| \geq 2(-1) + 1\)
  • Left side: \(|9 - 9| = |0| = 0\)
  • Right side: \(-2 + 1 = -1\)
  • Check: \(0 \geq -1\) ✓

• For \(\mathrm{x} = 4\), \(\mathrm{y} = 1\): \(|3(4) - 9| \geq 2(1) + 1\)
  • Left side: \(|12 - 9| = |3| = 3\)
  • Right side: \(2 + 1 = 3\)
  • Check: \(3 \geq 3\) ✓

Choice A passes all tests!

3. SIMPLIFY by checking the other choices for completeness

• Choice B fails immediately: \(\mathrm{x} = 2\), \(\mathrm{y} = 2\) gives \(3 \geq 5\), which is false
• Choice C fails at: \(\mathrm{x} = 3\), \(\mathrm{y} = 0\) gives \(0 \geq 1\), which is false
• Choice D fails at: \(\mathrm{x} = 3\), \(\mathrm{y} = 0\) gives \(0 \geq 1\), which is false

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make arithmetic errors when computing absolute values or combining terms, especially with negative numbers. For example, they might calculate \(|3(3) - 9|\) as 3 instead of 0, or compute \(2(-1) + 1\) as -3 instead of -1.

This leads to incorrect conclusions about which coordinate pairs satisfy the inequality, causing them to select Choice B, C, or D or abandon systematic checking and guess.

Second Most Common Error:

Poor TRANSLATE reasoning: Students may test only one or two coordinate pairs per table instead of checking all three pairs, thinking that passing some tests means the entire table works.

This incomplete approach may lead them to select Choice B, C, or D since partial testing might make these look viable.

The Bottom Line:

This problem requires methodical testing of every coordinate pair with careful arithmetic - there's no shortcut, and precision with absolute values and negative numbers is essential for success.