Problem 102
Question
Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation \(3(x+4)=3(4+x)\) has precisely one solution.
Step-by-Step Solution
Verified Answer
The statement 'The equation \(3(x+4)=3(4+x)\) has precisely one solution' is false. The corrected statement is 'The equation \(3(x+4)=3(4+x)\) has infinite solutions.'
1Step 1 - Analyze the equation
The equation given is \(3(x+4)=3(4+x)\) . The first step is to expand both sides of the equation by distributing 3 to \(x+4\) and \(4+x\).
2Step 2 - Simplify and compare
After distributing, the left side will be \(3x+12\) and the right side will be \(12+3x\) . If we rearrange the terms on the right hand side, we get \(3x+12\) which is same as left hand side. So, the equation \(3x+12 = 3x+12\) is always true for all values of \(x\). Hence, not only one solution exists.
3Step 3 - Correction of the statement
The original statement is false because it states there is precisely one solution. To correct the statement, it should state 'The equation \(3(x+4)=3(4+x)\) has infinite solutions.'
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