Intermediate Algebra Questions
With Solutions and Explanations - sample 3

Solutions with full explanations of the intermediate algebra questions in sample 3 are presented.

  1. Write 1.5 10-5 in standard form.

    Solution

    1.5 10-5 = 1.5 / 10 5 = 1.5 / 100000 = 0.000015

  2. Evaluate: 30 - |-x + 6| for x = 10

    Solution

    Substitute x by 10 in the given expression and evaluate

    30 - |- (10) + 6| = 30 - |-10 + 6|

    = 30 - | - 4 | = 30 - 4 = 26

  3. Evaluate: 2xy3 + x - 2y for x = 2 and y = -2

    Solution

    Substitute x by 2 and y by - 2 in the given expression and evaluate

    2(2)(-2)3 + 2 - 2(-2)

    = 4(-8) + 2 + 4 = - 26

  4. What is the slope of the line perpendicular to the line y = - 4

    Solution

    y = - 4 is a horizontal line and any line perpendicular to it is vertical and therefore has an undefined slope.

  5. Write an equation of the line with slope 2 and x-intercept (- 4 , 0).

    Solution

    A line with slope m and passes through point (a , b) has an equation of the form.

    y - b = m(x - a) , point slope form of a line

    Use the given slope and the point to write the equation of the line.

    y - 0 = 2(x - (-4))

    y = 2x + 8

  6. Solve the equation: -3(-x + 5) + 20 = -10(x - 3) + 4

    Solution

    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

  7. 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

  8. 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

  9. 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

  10. 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)

  11. 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)

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

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

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

  15. How many solutions does the equation 2x2 + 3x = 8 have?

  16. Solve the equation 3x2 + 6x - 1 = 8.

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

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

  19. Factor the expression 2x2 + 3x + 1.

  20. Factor the expression 10x2 + 20x - 80.







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