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 “no 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 “no solutions.”

-2x - 6(-x - 5) = 9x - 1 - 3

Explanation:

Problem 3

Step1: Distribute -7

Multiply -7 with each term inside the parentheses: $-7(-2x - 2) = 14x + 14$

Step2: Set up the equation

Now the equation is $14x + 14 = 35x + 40$

Step3: Subtract 14x from both sides

$14 = 21x + 40$

Step4: Subtract 40 from both sides

$-26 = 21x$

Step5: Divide by 21

$x = -\frac{26}{21}$ (Wait, the original handwritten answer was $x=\frac{26}{21}$, but let's check again. Wait, when we do $14x + 14 = 35x + 40$, subtract 14x: $14 = 21x + 40$, then subtract 40: $14 - 40 = 21x$ → $-26 = 21x$ → $x = -\frac{26}{21}$. Maybe the handwritten answer had a sign error. Let's redo the steps carefully.

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

Distribute left side: $(-7)(-2x) + (-7)(-2) = 14x + 14$

So equation: $14x + 14 = 35x + 40$

Subtract 14x from both sides: $14 = 21x + 40$

Subtract 40: $14 - 40 = 21x$ → $-26 = 21x$ → $x = -\frac{26}{21}$

But let's check the handwritten answer: $x=\frac{26}{21}$. Maybe a sign mistake. Let's confirm:

Left side: $-7(-2x - 2) = 14x + 14$

Right side: $35x + 40$

Bring all terms to left: $14x + 14 - 35x - 40 = 0$ → $-21x - 26 = 0$ → $-21x = 26$ → $x = -\frac{26}{21}$. So the correct solution is $x = -\frac{26}{21}$.

Step1: Distribute -6

Left side: $-2x - 6(-x - 5) = -2x + 6x + 30 = 4x + 30$

Step2: Simplify right side

Right side: $9x - 1 - 3 = 9x - 4$

Step3: Set up the equation

$4x + 30 = 9x - 4$

Step4: Subtract 4x from both sides

$30 = 5x - 4$

Step5: Add 4 to both sides

$34 = 5x$ (Wait, wait, original right side: $9x -1 -3 = 9x -4$. So equation: $4x + 30 = 9x -4$

Subtract 4x: $30 = 5x -4$

Add 4: $34 = 5x$ → $x = \frac{34}{5} = 6.8$? Wait, the handwritten answer was $x\frac{35}{5}$ (maybe a typo, maybe $x=\frac{35}{5}$ which is 7, but let's check again.

Wait, let's redo the left side: $-2x -6(-x -5) = -2x +6x +30 = 4x +30$

Right side: $9x -1 -3 = 9x -4$

So equation: $4x + 30 = 9x -4$

Subtract 4x: $30 = 5x -4$

Add 4: $34 = 5x$ → $x = \frac{34}{5} = 6.8$ or $x = 6\frac{4}{5}$. Wait, maybe I made a mistake in distribution. Wait, $-6(-x -5)$: $-6*(-x) = 6x$, $-6*(-5) = 30$, so left side is $-2x +6x +30 = 4x +30$. Correct.

Right side: $9x -1 -3 = 9x -4$. Correct.

So $4x +30 = 9x -4$

Subtract 4x: $30 = 5x -4$

Add 4: $34 = 5x$ → $x = \frac{34}{5}$. Wait, the handwritten answer was $x\frac{35}{5}$ (maybe a typo, maybe 35 instead of 34). Let's check if there was a miscalculation. Let's plug x=7 (which is 35/5) into the original equation.

Left side: $-2(7) -6(-7 -5) = -14 -6(-12) = -14 +72 = 58$

Right side: $9(7) -1 -3 = 63 -4 = 59$. Not equal. So x=7 is wrong.

Plug x=34/5=6.8:

Left side: $-2(34/5) -6(-34/5 -5) = -68/5 -6(-34/5 -25/5) = -68/5 -6(-59/5) = -68/5 + 354/5 = 286/5 = 57.2$

Right side: $9(34/5) -1 -3 = 306/5 -4 = 306/5 -20/5 = 286/5 = 57.2$. So correct. So the correct solution is $x = \frac{34}{5}$.

Wait, maybe the original problem had a different number? Let's check the original problem again: $-2x -6(-x -5) = 9x -1 -3$

Yes, that's what's written. So the correct solution is $x = \frac{34}{5}$.

Answer:

$x = -\frac{26}{21}$

Problem 4