Detailed solutions to algebra problems are presented.

__Solution to Problem 1:__

Given the equation

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

Multiply factors.

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

Group like terms.

- 16 x - 7 = - 16 x - 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) + 4 b - 2 (a - b - 3) + 5

Multiply factors.

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

Group like terms.

= 6 b + 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

2 x - 4 y = 9

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

2 x - 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) = 6 x + 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 = m x + 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.

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

Add 3 to both sides of the equation and simplify.

|-2 x + 2| = 0

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

x = 1

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