Expand the expressions with brackets in the given equation

3x - 15 + 20 = -10x + 30 + 4

Group like terms

3x + 5 = -10x + 34

Solve

13 x = 29 , x = 29 / 13

Solve the inequality: 4(x - 6) + 4 < 8(x - 4)

Solution

Expand the expressions with brackets in the given inequality

4x - 24 + 4 < 8x - 32

Group like terms

4x - 20 < 8x - 32

Solve the inequality

4x - 8x < -32 + 20

-4x < -12

x > 3

Solve the equation: 3(x - 2)^{2} - 12 = 0

Solution

Rewrite equation with square on one side

3(x - 2)^{2} = 12

(x - 2)^{2} = 4

x - 2 = (+ or -) 2

x = 2 + 2 = 4 , x = 2 - 2 = 0

The solutions of the given equation are

x = 4 and x = 0

Solve the equation: x / 3 + 2 / 7 = x / 7 - 5

Solution

The given equation has denominators that need to be cleared by multiplying all terms of the equation by the lowest common denominator of the denominators 3 and 7 which is 21

21(x / 3 + 2 / 7) = 21(x / 7 - 5)

Expand and simplify

7x + 6 = 3x - 105

Solve for x

4x = - 111 , x = - 111/4

Line L is defined as line through the point (2 , 7) and perpendicular to the line x + y = 0. What is the point of intersection of L and the line x + y = 0?

Solution

We first find the slope of the line x + y = 0. Rewrite in slope intercept form

y = - x , slope = - 1

Line L is perpendicular to line x + y = 0 and therefore the product of its slope m and the slope of the line x + y = 0 is equal to -1. Hence

m*(-1) = -1 , solve for m: m = 1

Let A(a , b) be the point of intersection. Point A and the given point (2 , 7) lie on line L and therefore the slope calculated using point (a , b) and (2 , 7) must be equal to the slope of line L which is - 1. Hence

(7 - b) / (2 - a) = 1

Point A(a , b) lie also on line x + y = 0. Hence

a + b = 0

Solve the two system of equations (7 - b) / (2 - a) = 1
and a + b = 0 to find point A(a , b).

a + b = 0 implies that a = - b

Cross multiply (7 - b) / (2 - a) = 1 to obtain

7 - b = 2 - a

Substitute a by - b (a = - b above) in the above equation and solve

7 - b = 2 - (-b) , b = 5 / 2

Use a = - b (see above) to find a

a = - 5/2

The point of intersection is

A(- 5/2 , 5/2)

What is the point of intersection of the lines: x + 2y = 4 and -x - 3y = -7?

Solution

Since the point of intersection lie on the two lines, its coordinates x and y satisfy the two equations simultaneously and are thefore found by solving the system of equations of the two lines. Let us add the right hand side and left hand side of the two equations

x + 2y = 4
+
- x - 3y = -7

- y = - 3 , equation obtained

y = 3

Substitute y by 3 in one of the equations to find x.

x + 2(3) = 4 , solve for x: x = - 2

The point of intersection is given by its coordinate as follows

(-2 , 3)

How many solutions do the system of eaquations 2x - 3y = 4 and 4x - 6y = -7 have?

For what value(s) of A does the system of equation Ax + 6y = 0 and 2x - 7y = 3 have no solutions?

Solve |2x - 4| - 2 = 6.

How many solutions does the equation 2x^{2} + 3x = 8 have?

Solve the equation 3x^{2} + 6x - 1 = 8.

Solve the system of equations: 2x + 5y = 18 and -3x - y = -1.

What is the range of function f defined by: f = {(2,3),(1,4),(5,4),(0,3)}