Detailed solutions to algebra problems are presented.

__Solution to Problem 1:__
Given the equation

5(-3x - 2) - (x - 3) = -4(4x + 5) + 13

Multiply factors.

-15x - 10 - x + 3 = -16x - 20 +13

Group like terms.

-16x - 7 = -16x - 7

Add 16x + 7 to both sides and write the equation as follows

0 = 0

The above statement is true for all values of x and therefore all real numbers are solutions to the given equation.

__Solution to Problem 2:__
Given the algebraic expression

2(a -3) + 4b - 2(a -b -3) + 5

Multiply factors.

= 2a - 6 + 4b -2a + 2b + 6 + 5

Group like terms.

= 6b + 5

__Solution to Problem 3:__
Given the expression

|x - 2| - 4|-6|

If x < 2 then x - 2 < 0 and if x - 2 < 0 then |x - 2| = -(x - 2).

Substitute |x - 2| by -(x - 2) and |-6| by 6 .

|x - 2| - 4|-6| = - (x - 2) - 4(6) = - x -22

__Solution to Problem 4:__
The distance d between points (-4 , -5) and (-1 , -1) is given by

d = √[ (-1 - (-4))^{ 2} + (-1 - (-5))^{ 2} ]

Simplify.

d = √(9 + 16) = 5

__Solution to Problem 5:__
Given the equation

2x - 4y = 9

To find the x intercept we set y = 0 and solve for x.

2x - 0 = 9

Solve for x.

x = 9 / 2

The x intercept is at the point (9/2 , 0).

__Solution to Problem 6:__
Given the function

f(x) = 6x + 1

f(2) - f(1) is given by.

f(2) - f(1) = (6*2 + 1) - (6*1 + 1) = 6

__Solution to Problem 7:__
Given the points (-1, -1) and (2 , 2), the slope m is given by

m = (y2 - y1) / (x2 - x1) = (2 - (-1)) / (2 - (-1)) = 1

__Solution to Problem 8:__
Given the line

5x - 5y = 7

Rewrite the equation in slope intercept form y = mx + b and identify the value of m the slope.

-5y = -5x + 7

y = x - 7/5

The slope is given by the coefficient of x which is 1.

__Solution to Problem 9:__
To find the equation of the line through the points (-1 , -1) and (-1 , 2), we first use the slope m.

m = (y2 - y1) / (x2 - x1) = (2 - (-1)) / (-1 - (-1)) = 3 / 0

The slope is undefined which means the line is perpendicular to the x axis and its equation has the form x = constant. Since both points have equal x coordinates -1, the equation is given by:

x = -1

__Solution to Problem 10:__
The equation to solve is given by.

|-2x + 2| -3 = -3

Add 3 to both sides of the equation and simplify.

|-2x + 2| = 0

|-2x + 2| is equal to 0 if -2x + 2 = 0. Solve for x to obtain

x = 1
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