Question

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1. when solving an equation, david’s first step is shown below. which property justifies david’s first step?

original equation:
$-4x + x^2 - 1 = 0$
first step:
$x^2 - 4x - 1 = 0$
a. commutative property of addition
b. division property of equality
c. commutative property of multiplication
d. associative property of multiplication

1. when solving an equation, arturo’s first step is shown below. which property justifies arturo’s first step?

original equation:
$-6(-9x^2 - 9) + 9 = 9x^2 + 9$
first step:
$-6(-9x^2 - 9) = 9x^2$
a. associative property of multiplication
b. commutative property of addition
c. distributive property of multiplication over addition
d. subtraction property of equality

1. solve for $x$:

$3x + 5 = 5x + 25$
name________ period__

1. solve: $5(x - 4) = 15$
2. solve for $x$ and graph the solution on the number line below.

$-6x > -12$
(number line image: -12, -10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10, 12 with arrows)

1. solve the inequality and graph the solution on the line provided.

$-18 + 5x > 12$
(number line image: -12, -10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10, 12 with arrows)

Explanation:

Question 1

Step1: Recall properties

The commutative property of addition states that \(a + b = b + a\) (order of addition doesn't matter). The original equation is \(-4x + x^2 - 1 = 0\), and the first step is \(x^2 - 4x - 1 = 0\). Here, the terms \(-4x\) and \(x^2\) were reordered, which is using the commutative property of addition. Division property of equality involves dividing both sides, multiplication commutative is about \(ab = ba\), and associative multiplication is about \((ab)c = a(bc)\), which don't apply here.

Step1: Analyze the step

Original equation: \(-6(-9x^2 - 9)+9 = 9x^2 + 9\). First step: \(-6(-9x^2 - 9)=9x^2\). This is subtracting 9 from both sides, which is the subtraction property of equality (if \(a + b = c\), then \(a = c - b\)). Associative multiplication, commutative addition, and distributive property don't fit here.

Step1: Subtract \(3x\) from both sides

\(3x + 5 - 3x=5x + 25 - 3x\)
Simplify: \(5 = 2x + 25\)

Step2: Subtract 25 from both sides

\(5 - 25=2x + 25 - 25\)
Simplify: \(-20 = 2x\)

Step3: Divide by 2

\(\frac{-20}{2}=\frac{2x}{2}\)
Simplify: \(x=-10\)

Answer:

A. commutative property of addition

Question 2