Which of the following expressions is equivalent to ?
For expression-equivalence questions, expand step by step and combine like terms carefully. Distribute coefficients (including negative signs) to every term inside parentheses, then group and add the -terms and constant terms separately. Keeping track of signs is crucial; a quick mental check of one or two -values (plugging into both the original and a candidate expression) can also confirm whether an option is equivalent without redoing all the algebra.
Hints
Focus on distribution
First, expand and by multiplying the number outside each set of parentheses by both terms inside.
Be careful with the negative sign
When you expand , think of it as multiplying by both and . What signs should the resulting terms have?
Combine like terms
After you expand, you should have two -terms and two constant terms. Add the -terms together and the constants together to get a single simplified expression.
Desmos Guide
Enter the original expression
In the first Desmos line, type 5(x+4)-3(x-2) to represent the given expression.
Enter each answer choice as a separate expression
On new lines, type each option exactly as written: 2x+14, 8x-8, 8x+14, and 2x+26. You will now have five graphs or expressions listed.
Compare graphs or values
Look at the graphs: the correct choice will have a line that lies exactly on top of the graph of 5(x+4)-3(x-2) for all . Alternatively, use a table (click the gear icon and add a table) to check that the -values match for several -values; the option that always matches is the equivalent expression.
Step-by-step Explanation
Distribute each factor over its parentheses
Start by distributing the numbers outside the parentheses:
- For , multiply 5 by both and 4.
- For , remember the negative sign belongs to the 3, so you are really distributing .
After distributing, rewrite the expression using your results.
Write the expanded expression
Compute each product from the distribution:
- becomes .
- becomes (because and ).
So the original expression becomes:
Combine like terms to simplify fully
Group the -terms together and the constant terms together:
So the simplified expression is , which corresponds to choice D.