Select all expressions that are equivalent to 6x +1 – (3x – 1) Hint: Don’t forget to distribute the negative! 6x +1 -3.2 – 1 6x + (-3x) +1+1 3x + 2 6x – 3x + 1 – 1 6x +1+(-3x) – (-1)
The Correct Answer and Explanation is:
Let’s solve this step by step and check which expressions are equivalent to 6x + 1 – (3x – 1).
Step 1: Simplify the Original Expression
The expression is:
6x + 1 – (3x – 1)
Distribute the negative sign:
6x + 1 – 3x + 1
Combine like terms:
6x – 3x + 1 + 1 = 3x + 2
So, the simplified form is:
3x + 2
Step 2: Check Each Option
Option 1:
6x + 1 – 3.2 – 1
Notice the term 3.2, which is a decimal. Our expression uses 3x, not 3.2, so this is not equivalent.
Option 2:
6x + (-3x) + 1 + 1
Simplify:
6x – 3x + 1 + 1 = 3x + 2
This matches the simplified form, so this is equivalent.
Option 3:
3x + 2
This is the same as our simplified form, so this is equivalent.
Option 4:
6x – 3x + 1 – 1
Simplify:
6x – 3x + 1 – 1 = 3x + 0 = 3x
This does not match 3x + 2, so it is not equivalent.
Option 5:
6x + 1 + (-3x) – (-1)
Simplify step by step:
6x + 1 – 3x + 1 = 3x + 2
This matches the simplified form, so this is equivalent.
Final Answer
The expressions that are equivalent to 6x + 1 – (3x – 1) are:
✔ 6x + (-3x) + 1 + 1
✔ 3x + 2
✔ 6x + 1 + (-3x) – (-1)
Explanation
When simplifying algebraic expressions, it is essential to handle negative signs properly, especially when parentheses are involved. In the given expression 6x + 1 – (3x – 1), the parentheses mean that everything inside must be affected by the negative sign placed in front. Distributing the negative sign changes 3x – 1 to -3x + 1.
This results in 6x + 1 – 3x + 1, which simplifies by combining like terms. The terms involving x are 6x and -3x, which add to 3x. The constant terms 1 and 1 add to 2, giving the simplified expression 3x + 2.
To determine which given expressions are equivalent, each option must be simplified and compared to 3x + 2. Expressions that involve decimals like 3.2 are immediately ruled out because they differ in form and value.
Correct expressions use proper distribution and grouping of terms. For example, 6x + (-3x) + 1 + 1 simplifies directly to 3x + 2, matching the original simplified form. Similarly, 6x + 1 + (-3x) – (-1) becomes 3x + 2 after simplifying both the variables and constants.
Incorrect expressions fail to simplify to 3x + 2. For instance, 6x – 3x + 1 – 1 simplifies to 3x, missing the constant term +2, making it non-equivalent.
Thus, understanding distribution and combining like terms is crucial to finding equivalent expressions accurately.
