If one fourth of x minus four fifths of six sevenths is equal to -9/7, then what is the exact value of x? Write the equation carefully as (1/4)*x - (4/5)*(6/7) = -9/7 and solve for x without approximation.

Introduction / Context:
This simplification question checks whether you can translate words into a correct algebraic equation and then solve a linear equation involving fractions. Many errors happen because students mix up “of” (meaning multiplication) or subtract the wrong part. Here, “one fourth of x” means (1/4)*x, and “four fifths of six sevenths” means (4/5)*(6/7). Once you convert the sentence correctly, solving is just careful fraction arithmetic.

Given Data / Assumptions:

• (1/4)*x - (4/5)*(6/7) = -9/7 • Required: value of x • Use exact fraction operations (no rounding needed)

Concept / Approach:
Compute the constant product (4/5)*(6/7) first. Then move it to the other side to isolate x/4. Finally multiply both sides by 4 to get x. Keeping a common denominator (like 35) helps avoid mistakes.

Step-by-Step Solution:
1) Start with the equation: (1/4)*x - (4/5)*(6/7) = -9/7 2) Compute the product: (4/5)*(6/7) = 24/35 3) Rewrite -9/7 with denominator 35: -9/7 = -45/35 4) Add 24/35 to both sides: x/4 = (-45/35) + (24/35) = -21/35 5) Simplify -21/35: -21/35 = -3/5 6) Multiply both sides by 4: x = 4 * (-3/5) = -12/5 = -2.4

Verification / Alternative check:
Substitute x = -12/5 into (1/4)*x: (1/4)*(-12/5) = -3/5. Convert -3/5 to -21/35 and subtract 24/35 to get (-21/35) - (24/35) = -45/35 = -9/7, which matches the right side. So the solution is consistent.

Why Other Options Are Wrong:
• -3.6 and -1.8: come from incorrect addition/subtraction of the fractions. • -12 and -14: usually happen when multiplying by 4 at the wrong time or misreading “one fourth of x.”

Common Pitfalls:
• Treating “one fourth of x” as x - 1/4 instead of (1/4)*x. • Forgetting that “of” means multiplication and not division. • Mixing denominators without converting to a common denominator.

Final Answer:
-2.4