Question

heather is solving an inequality. she uses the following steps to reach her solution.
which of the following is an error that heather made in reaching her solution?
a simplifying error between line 4 and line 5.
b distributive error between line 2 and line 3.
c combination of fractions error between line 3
and line 4.
d failure to swap the inequality sign between line

Explanation:

To solve this, we analyze each option by recalling inequality-solving steps (distribution, combining fractions, simplifying, and inequality sign rules):

Option A: Simplifying error between Line 4 and Line 5

Simplifying errors involve miscalculating arithmetic (e.g., adding/subtracting terms incorrectly). Without seeing Line 4–5, we first check other options.

Option B: Distributive error between Line 2 and Line 3

The distributive property is \( a(b + c) = ab + ac \). If Line 2 uses a distributive step (e.g., multiplying a fraction across parentheses) and Line 3 misapplies it (e.g., incorrect sign or coefficient), this is a common error. For example, if distributing \( \frac{1}{2}(x + 3) \), it should be \( \frac{1}{2}x + \frac{3}{2} \); a mistake here would be a distributive error.

Option C: Combination of fractions error between Line 3 and Line 4

Combining fractions requires a common denominator. If Line 3 has fractions and Line 4 miscombines them (e.g., adding numerators without common denominators), but this is less likely than a distributive error (which is a foundational step).

Option D: Failure to swap the inequality sign

The inequality sign is only swapped when multiplying/dividing by a negative number. If no negative multiplication/division occurred, this error is irrelevant.

Distributive errors are common in early steps (Line 2–3) when expanding parentheses. Thus, the most probable error is a distributive error between Line 2 and Line 3.

Answer:

B. Distributive error between Line 2 and Line 3.