Problem 153 - Algebra and Trigonometry

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$5+3(x-4)=8(x-4)=8 x-32$$

Step-by-Step Solution

Answer

The given statement is partially true. The two terms on the RHS of the equal signs are equal, but the item on the LHS of the first equation does not equal to the others. Thus, the correct statement should be \(3x-7=3x-7\).

Step 1: Evaluate LHS of first equation

Let's evaluate the left-hand side (LHS) of the first equation which is \(5+3(x-4)\). This simplifies to: \(5 + 3x - 12\) which further simplifies to: \(3x - 7\)

Step 2: Evaluate RHS of first equation

Now, let's evaluate the right-hand side (RHS) of the first equation which is \(8(x-4)\). This simplifies to: \(8x - 32\)

Step 3: Compare LHS and RHS of the first equation

Comparing the LHS (\(3x - 7\)) with the RHS (\(8x - 32\)) it is clear that they are not equal, therefore the first equation is False.

Step 4: Evaluate RHS of second equation

Let's evaluate the right-hand side (RHS) of the second equation which is \(8x - 32\). This value is already simplified.

Step 5: Compare RHS of first and second equations

Comparing the RHS of the first equation (\(8x - 32\)) with the RHS of the second equation (\(8x - 32\)): They are equal, so the second equation is True.

Step 6: Provide the corrected equation

Since the LHS of the first equation does not match the RHS, a corrected statement would be: \(3x-7=3x-7\)