During the restoration of an ancient bridge, Emek and Tara assisted engineers by reviewing inspection records containing algebraic fractions for each structural checkpoint. They simplified every expression, confirmed that each result satisfied the required restrictions on variable values, and compared their work with the recorded calculations. One checkpoint contained an incorrect simplification caused by canceling terms improperly instead of canceling common factors. After correcting the mistake and verifying the remaining records, they completed the inspection with confidence. The bridge passed its final safety review and was reopened for travelers.
Challenges
Challenge 1 of 5
Challenges
Challenge 1 of 4
Hint
A common factor may be canceled, but the original denominator cannot be zero.
During the bridge inspection, one engineer simplified (x + 4)/(x + 4) to 1. Which statement is correct?
The simplification is correct for all real numbers.The simplification is correct only when x ≠ -4.The expression simplifies to x.The expression cannot be simplified.Correct! The expression simplifies to 1, but x = -4 is not allowed.
Remember to keep the restriction from the original denominator.
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Hint
Find which value makes the denominator zero.
One inspection record contains the expression (x² - 49)/(x - 7). How many integer values of x from 5 through 9 are not allowed because the original denominator equals zero?
Correct! Only one value is excluded.
Check when x - 7 equals zero.
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Hint
Think about the difference between factors and terms.
An engineer incorrectly simplified (x + 6)/(x + 2) by canceling the '+2' and writing the result as 3. Why is this simplification incorrect?
The denominator should be multiplied first.Terms in a sum cannot be canceled; only common factors can be canceled.The numerator should be squared before simplifying.The expression should always equal 1.Correct! Only common factors may be canceled, not individual terms in addition or subtraction.
Review the rule that cancellation is only valid for common factors.
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