Question

1. solve the following equation for ( x ). express your answer in the simplest form. if there are infinite solutions state \infinite solutions\ and if there are no solutions state

o solutions.\
( -7(-2x - 2) = 35x + 40 )

1. solve the following equation for ( x ). express your answer in the simplest form. if there are infinite solutions state \infinite solutions\ and if there are no solutions state

o solutions.\
( -2x - 6(-x - 5) = 9x - 1 - 3 )

Explanation:

Problem 3:

Step 1: Distribute the -7

We start by applying the distributive property to the left side of the equation \(-7(-2x - 2)\). The distributive property states that \(a(b + c)=ab+ac\), so here \(a = - 7\), \(b=-2x\) and \(c = - 2\).
\[

$$\begin{align*} -7(-2x-2)&=(-7)\times(-2x)+(-7)\times(-2)\\ &=14x + 14 \end{align*}$$

\]
So the equation becomes \(14x + 14=35x + 40\).

Step 2: Move all x terms to one side

Subtract \(14x\) from both sides of the equation:
\[
14x+14 - 14x=35x + 40-14x
\]
Simplifying both sides, we get \(14 = 21x+40\).

Step 3: Move constant terms to the other side

Subtract 40 from both sides:
\[
14-40=21x + 40-40
\]
Simplifying, we have \(- 26=21x\) (Wait, there was a mistake in the original hand - written answer. Let's correct it). Wait, \(14 - 40=-26\), so \(-26 = 21x\), then \(x=-\frac{26}{21}=-\frac{2\times13}{3\times7}\), but let's re - do the steps correctly.

Wait, original equation: \(-7(-2x - 2)=35x + 40\)

Step 1: Distribute \(-7\): \(14x+14 = 35x+40\)

Step 2: Subtract \(14x\) from both sides: \(14=21x + 40\)

Step 3: Subtract 40 from both sides: \(14 - 40=21x\)

\(-26 = 21x\)

\(x=-\frac{26}{21}\)

Wait, the hand - written answer was \(x = \frac{26}{21}\) with a negative sign missing. Let's do it again:

Starting over:

Equation: \(-7(-2x - 2)=35x + 40\)

Distribute left side: \(14x + 14=35x+40\)

Subtract \(35x\) from both sides: \(14x-35x + 14=35x-35x + 40\)

\(-21x+14 = 40\)

Subtract 14 from both sides: \(-21x+14 - 14=40 - 14\)

\(-21x=26\)

Divide both sides by \(-21\): \(x=-\frac{26}{21}\)

Problem 4:

Step 1: Distribute the -6

We apply the distributive property to the left side of the equation \(-2x-6(-x - 5)\). Using \(a(b + c)=ab+ac\) with \(a=-6\), \(b=-x\) and \(c=-5\):
\[

$$\begin{align*} -2x-6(-x - 5)&=-2x+(-6)\times(-x)+(-6)\times(-5)\\ &=-2x + 6x+30 \end{align*}$$

\]
Simplify the left side: \((-2x + 6x)+30 = 4x+30\)

Simplify the right side: \(9x-1 - 3=9x-4\)

So the equation becomes \(4x + 30=9x-4\)

Step 2: Move x terms to one side

Subtract \(4x\) from both sides:
\[
4x+30 - 4x=9x-4 - 4x
\]
Simplifying, we get \(30 = 5x-4\)

Step 3: Move constant terms to the other side

Add 4 to both sides:
\[
30 + 4=5x-4 + 4
\]
Simplifying, we have \(34 = 5x\) (Wait, the hand - written answer was \(x=\frac{35}{5}\), which is wrong. Let's do it correctly)

Wait, original equation: \(-2x-6(-x - 5)=9x-1 - 3\)

Left side after distribution: \(-2x + 6x+30=4x + 30\)

Right side: \(9x-4\)

Equation: \(4x + 30=9x-4\)

Subtract \(4x\) from both sides: \(30 = 5x-4\)

Add 4 to both sides: \(30 + 4=5x\)

\(34 = 5x\)

\(x=\frac{34}{5}=6.8\)

Wait, let's re - check the right side: \(9x-1 - 3=9x-(1 + 3)=9x - 4\), correct. Left side: \(-2x-6(-x - 5)=-2x + 6x + 30=4x + 30\), correct. Then \(4x+30=9x - 4\), subtract \(4x\): \(30 = 5x-4\), add 4: \(34 = 5x\), so \(x=\frac{34}{5}\)

Correct Answers:

Problem 3:

Step 1: Distribute \(-7\)

Using the distributive property \(a(b + c)=ab+ac\) on \(-7(-2x-2)\), we get \(14x + 14=35x + 40\).

Step 2: Subtract \(14x\) from both sides

\(14=21x + 40\)

Step 3: Subtract 40 from both sides

\(-26 = 21x\)

Step 4: Solve for \(x\)

\(x=-\frac{26}{21}\)

Answer:

\(x = -\frac{26}{21}\)

Problem 4: