Question

free response questions

1. solve the following equation for v in terms of all other variables involved.

\\( \frac { 2 ( v - h ) } { k } = r \\)

1. give a property of real numbers (associative, commutative, or distributive) or a property of equality (addition or multiplication) that justifies each step in the solution of the equation shown below.

\\( 5 ( x + 3 ) + 2 x = 4 x + 9 \\)
(1) \\( 5 x + 15 + 2 x = 4 x + 9 \\)
(2) \\( 5 x + 2 x + 15 = 4 x + 9 \\)
(3) \\( ( 5 + 2 ) x + 15 = 4 x + 9 \\)
\\( 7 x + 15 = 4 x + 9 \\)
(4) \\( 7 x + 15 - 15 = 4 x + 9 - 15 \\)
\\( 7 x = 4 x - 6 \\)
(5) \\( 7 x - 4 x = 4 x - 6 - 4 x \\)
(6) \\( ( 7 - 4 ) x = - 6 \\)
\\( 3 x = - 6 \\)
(7) \\( \frac { 1 } { 3 } cdot 3 x = \frac { 1 } { 3 } cdot - 6 \\)
\\( x = - 2 \\)

1. solve the inequality shown below. show all steps. graph the solution set on the number line provided.

\\( 5 ( 2 x + 1 ) - 6 ( x + 1 ) > 11 x - 29 \\)

Explanation:

Step1: Expand the brackets

$$\begin{align*} 5(2x + 1)-6(x + 1)&>11x-29\\ 10x+5-(6x + 6)&>11x-29\\ 10x+5-6x-6&>11x-29 \end{align*}$$

Step2: Combine like terms

$$\begin{align*} (10x-6x)+(5 - 6)&>11x-29\\ 4x-1&>11x-29 \end{align*}$$

Step3: Subtract \(4x\) from both sides

$$\begin{align*} 4x-1-4x&>11x-29-4x\\ -1&>7x-29 \end{align*}$$

Step4: Add \(29\) to both sides

$$\begin{align*} -1 + 29&>7x-29+29\\ 28&>7x \end{align*}$$

Step5: Divide both sides by \(7\)

$$\begin{align*} \frac{28}{7}&>\frac{7x}{7}\\ 4&>x \end{align*}$$

Answer:

The solution of the inequality is \(x < 4\). On the number - line, we draw an open circle at \(x = 4\) (because the inequality is strict, \(x
eq4\)) and shade the part of the number - line to the left of \(4\).