Certainly! Let's break down the steps and name the property that justifies each step:
1. Starting Expression:
[tex]\[
(-2)\left(-\frac{1}{2} x\right)+3 x
\][/tex]
2. Step 1: Distribute [tex]\(-2\)[/tex] across the term inside the parenthesis [tex]\( \left(-2 \left( -\frac{1}{2} \right)\right) \)[/tex]:
[tex]\[
(-2)\left(-\frac{1}{2} x\right)+3 x = \left[(-2)\left(-\frac{1}{2}\right)\right] x + 3 x
\][/tex]
Property Justification: Multiplicative inverse property. We are distributing the multiplication over the subtraction inside the parenthesis.
3. Step 2: Simplify the multiplication inside the brackets:
[tex]\[
(-2)\left(-\frac{1}{2}\right) = 1
\][/tex]
Hence:
[tex]\[
1 \cdot x + 3 x
\][/tex]
Property Justification: Simplification. The product [tex]\((-2) \cdot \left(-\frac{1}{2}\right)\)[/tex] simplifies to [tex]\(1\)[/tex].
4. Step 3: Combine like terms using the associative property of addition:
[tex]\[
1x + 3x = (1+3)x
\][/tex]
Property Justification: Additive property. We are adding like terms (coefficients of [tex]\( x \)[/tex]).
5. Step 4: Final simplification:
[tex]\[
(1+3)x = 4x
\][/tex]
Property Justification: Simplification. Adding the coefficients inside the parenthesis.
So, the properties that justify each step in the solution are:
1. Multiplicative inverse property
2. Simplification
3. Additive property
4. Simplification
These steps explain how we reach from the initial expression to the final simplified form [tex]\(4x\)[/tex].
