Question

assessment practice
3a select each expression that is equivalent to
8.5 + (2s + 0.5)
(8.5 + 2s) + 0.5
(8.5 + 0.5) + 2s
9 + 2
2(4.5 + s)
8.5(2s + 0.5)

Explanation:

Step1: Recall the Associative Property of Addition

The associative property of addition states that for any real numbers \(a\), \(b\), and \(c\), \((a + b)+c=a+(b + c)\). Also, recall the commutative property of addition \(a + b=b + a\) and the distributive property \(a(b + c)=ab+ac\). We will check each option against the original expression \(8.5+(2s + 0.5)\).

Step2: Check \((8.5 + 2s)+0.5\)

Using the associative property of addition, \(a+(b + c)=(a + b)+c\) where \(a = 8.5\), \(b=2s\), \(c = 0.5\). So \(8.5+(2s + 0.5)=(8.5 + 2s)+0.5\). This expression is equivalent.

Step3: Check \((8.5 + 0.5)+2s\)

Using the commutative property of addition (to re - order \(2s\) and \(0.5\)) and then the associative property. First, by commutative property \(8.5+(2s + 0.5)=8.5+(0.5 + 2s)\), then by associative property \((8.5 + 0.5)+2s\). This expression is equivalent.

Step4: Check \(9\times2\)

Simplify the original expression: \(8.5+(2s+0.5)=(8.5 + 0.5)+2s=9 + 2s\). \(9\times2 = 18\), and \(9 + 2s\) is not equal to \(18\) (unless \(s=\frac{9}{2}\), but we are looking for equivalent expressions for all \(s\)). So this is not equivalent.

Step5: Check \(2(4.5 + s)\)

Expand \(2(4.5 + s)\) using the distributive property: \(2\times4.5+2\times s=9 + 2s\). The original expression simplifies to \(9 + 2s\) (from Step 4), so \(2(4.5 + s)=9 + 2s\) and is equivalent.

Step6: Check \(8.5(2s + 0.5)\)

Expand \(8.5(2s+0.5)\) using the distributive property: \(8.5\times2s+8.5\times0.5 = 17s+4.25\). The original expression is \(9 + 2s\), which is not equal to \(17s + 4.25\) for all \(s\). So this is not equivalent.

Answer:

\((8.5 + 2s)+0.5\), \((8.5 + 0.5)+2s\), \(2(4.5 + s)\)