Question

1. simplify: -2x + 11 + 6x
2. simplify: -8x - 11x
3. solve for x:

x - (-8) = 9

1. multiply:

(-8)(-2) =

1. solve for x:

x + 9.3 = 21.1

1. solve for x:

\frac{x}{4} = -11
extended response

1. find the perimeter of the rectangle below:

3x - 5
3x + 3

Explanation:

1) Simplify: $-2x + 11 + 6x$

Step1: Combine like - terms

Combine the $x$ - terms: $(-2x+6x)+11$.
Using the rule $ax + bx=(a + b)x$, we have $(-2 + 6)x+11$.

2) Simplify: $-8x-11x$

Step1: Combine like - terms

Using the rule $ax+bx=(a + b)x$ (here $a=-8$ and $b = - 11$), we get $(-8-11)x$.

3) Solve for $x$: $x-(-8)=9$

Step1: Simplify the left - hand side

Using the rule $a-(-b)=a + b$, the equation becomes $x + 8=9$.

Step2: Isolate $x$

Subtract 8 from both sides: $x+8 - 8=9 - 8$.

4) Multiply: $(-8)(-2)$

Step1: Apply the rule for multiplying two negative numbers

The rule is $(-a)(-b)=ab$ (where $a = 8$ and $b = 2$).

5) Solve for $x$: $x + 9.3=21.1$

Step1: Isolate $x$

Subtract 9.3 from both sides: $x+9.3-9.3=21.1 - 9.3$.

6) Solve for $x$: $\frac{x}{4}=-11$

Step1: Isolate $x$

Multiply both sides by 4: $\frac{x}{4}\times4=-11\times4$.

7) Find the perimeter of the rectangle

Step1: Recall the formula for the perimeter of a rectangle

The formula for the perimeter of a rectangle is $P = 2(l + w)$, where $l$ is the length and $w$ is the width. Here, $l=3x + 3$ and $w=3x - 5$.

Step2: Substitute into the formula

$P=2((3x + 3)+(3x - 5))$.
First, simplify the expression inside the parentheses: $(3x+3)+(3x - 5)=3x+3x+3 - 5=(3 + 3)x+(3 - 5)=6x-2$.
Then, $P=2(6x - 2)$.
Using the distributive property $a(b + c)=ab+ac$ (here $a = 2$, $b = 6x$, $c=-2$), we get $P=2\times6x-2\times2$.

Answer:

1. $4x + 11$
2. $-19x$
3. $x = 1$
4. $16$
5. $x=11.8$
6. $x=-44$
7. $12x-4$