Parentheses Removal and Changing Terms

Removing Parentheses: Positive Precursor

The Unchanged Pass-Through

When simplifying an expression, we can remove parentheses directly if they are preceded by a positive sign (++). The signs of all the terms inside the parentheses remain completely unchanged.
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Key Points

  1. Parentheses can be removed directly if preceded by a plus sign (++).
  2. A plus sign outside does not alter any internal signs.
  3. Example: 28+(35−10)28 + (35 - 10) simplifies directly to 28+35−1028 + 35 - 10.
  4. The signs of 3535 (positive) and −10-10 (negative) remain the same.

Olympiad Fact

This rule is a direct result of the identity property of +1+1 multiplication: multiplying any term by +1+1 preserves its original sign and value.

Removing Parentheses: Negative Precursor

The Sign Inversion Rule

If parentheses are preceded by a negative sign (−-), removing them requires reversing the sign of every term inside: positive terms become negative, and negative terms become positive.
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Key Points

  1. A minus sign (−-) outside parentheses acts as an inverter.
  2. Removing the parentheses requires reversing the sign of every internal term.
  3. Example: 84−(29−11)84 - (29 - 11) simplifies to 84−29+1184 - 29 + 11.
  4. The positive 2929 becomes −29-29, and the negative −11-11 becomes +11+11.

Olympiad Fact

This inversion happens because a negative sign outside a bracket represents multiplying each term inside by −1-1.

Effect of Changing Terms

Symmetrical Value Shifts

The overall value of an arithmetic expression is directly tied to the values of its individual terms. Any increase or decrease in a term produces a matching change in the final value.
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Key Points

  1. The value of an expression depends on the size of its individual terms.
  2. Increasing any term increases the final value by that same amount.
  3. Decreasing any term decreases the final value by that same amount.
  4. Example: If 43+(−16)=2743 + (-16) = 27, increasing 4343 by 11 to 4444 increases the sum to 2828.

Olympiad Fact

This direct relationship allows you to solve adjacent comparison questions instantly without recalculating the entire expression from scratch.

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