Q. negative plus a positive equals

Problem: “negative plus a positive” — step-by-step explanation

We will explain, in full detail, how to evaluate a sum in which one addend is negative and the other is positive. Write the negative number as \(-a\) with \(a>0\), and the positive number as \(b\) with \(b>0\).

1. Rewrite the sum in a subtraction form.

   The expression \(-a + b\) can be rewritten by moving terms so that the positive quantity comes first:

   \[ -a + b = b – a. \]

   Explanation: adding a negative number is the same as subtracting its absolute value. So \(-a + b\) is the same operation as subtracting \(a\) from \(b\).

2. Compare the absolute values.

   To determine the sign of the result, compare \(a\) and \(b\):

   • If \(b > a\), the positive number has the larger absolute value, so the result is positive.
   • If \(b < a\), the negative number (in absolute value \(a\)) is larger, so the result is negative.
   • If \(b = a\), the numbers cancel and the result is zero.
3. Compute the numerical result by subtracting the smaller absolute value from the larger, and attach the sign of the larger.

   Using the comparison above, compute the difference and determine the sign:

   • Case \(b > a\): \[ -a + b = b – a \quad\text{(positive result)}. \]
   • Case \(b < a\): \[ -a + b = -(a - b) \quad\text{(negative result)}. \]
   • Case \(b = a\): \[ -a + b = 0. \]
4. Worked examples (each shown step-by-step).
   1. Example 1: Evaluate \(-3 + 5\).

      Step 1: Rewrite as subtraction: \[ -3 + 5 = 5 – 3. \]

      Step 2: Compare absolute values: \(5 > 3\), so result is positive.

      Step 3: Subtract: \[ 5 – 3 = 2. \]

      Answer: \[ -3 + 5 = 2. \]

   2. Example 2: Evaluate \(-7 + 4\).

      Step 1: Rewrite: \[ -7 + 4 = 4 – 7. \]

      Step 2: Compare absolute values: \(7 > 4\), so result is negative.

      Step 3: Subtract the smaller from the larger and attach the negative sign: \[ 7 – 4 = 3 \Rightarrow – (7 – 4) = -3. \]

      Answer: \[ -7 + 4 = -3. \]

   3. Example 3: Evaluate \(-6 + 6\).

      Step 1: Rewrite: \[ -6 + 6 = 6 – 6. \]

      Step 2: Compare absolute values: \(6 = 6\), so they cancel.

      Step 3: Subtract: \[ 6 – 6 = 0. \]

      Answer: \[ -6 + 6 = 0. \]

5. Summary (rule to apply every time).

   To add a negative and a positive:

   1. Rewrite as a subtraction: \(-a + b = b – a\).
   2. Compare absolute values \(a\) and \(b\).
   3. Subtract the smaller absolute value from the larger.
   4. The sign of the result is the sign of the number with the larger absolute value; if equal, the result is zero.