Algebra

(3x - 1)/2 + 4 = (x + 5)/3

Step-by-step solution with explanation

Final Answer

x = -1

Step-by-step solution

Multiply every term by the LCD

The denominators are 2 and 3, so the least common denominator (LCD) is 6. Multiply every term on both sides by 6 to clear all fractions.

Simplify each term

6 divided by 2 is 3, and 6 divided by 3 is 2. This removes all fractions and makes the equation much easier to work with.

Distribute on both sides

Multiply 3 through (3x - 1) to get 9x - 3, and multiply 2 through (x + 5) to get 2x + 10.

Combine like terms on the left

-3 + 24 = 21, so the left side simplifies to 9x + 21.

Move x terms to one side

Subtract 2x from both sides. 9x - 2x = 7x, giving us 7x + 21 = 10.

Isolate x and solve

Wait — let's recheck: subtract 21 from both sides: 7x = 10 - 21 = -11, so x = -11/7. Let me verify by re-examining step 4.

Verify the answer by substituting back

and ✓

Both sides equal 8/7, confirming x = -11/7 is correct. The final answer is x = -11/7, not -1.

Understanding this problem

Learning Insight

Multiplying every term by the LCD is the key move for equations with fractions — it transforms a messy problem into a simple linear equation. This works because multiplying both sides of an equation by the same non-zero number keeps the equation balanced.

Quick Tip

To find the LCD quickly, list multiples of each denominator until you find the first one they share. For 2 and 3, multiples of 3 are 3, 6 — and 6 is also a multiple of 2, so LCD = 6.

Common Mistake

A very common mistake is forgetting to multiply the whole-number term (the +4 here) by the LCD. Students often only multiply the fraction terms, which gives a wrong equation.