Jay Abramson

Review Exercises

The Rectangular Coordinate Systems and Graphs

For the following exercises, find the x-intercept and the y-intercept without graphing.

1.

$4 x - 3 y = 12$

2.

$2 y - 4 = 3 x$

For the following exercises, solve for y in terms of x, putting the equation in slope–intercept form.

3.

$5 x = 3 y - 12$

4.

$2 x - 5 y = 7$

For the following exercises, find the distance between the two points.

5.

$(−2, 5) (4, −1)$

6.

$(−12, −3) (−1, 5)$

7.

Find the distance between the two points $(−71,432)$ and $(511,218)$ using your calculator, and round your answer to the nearest thousandth.

For the following exercises, find the coordinates of the midpoint of the line segment that joins the two given points.

8.

$(−1, 5)$ and $(4, 6)$

9.

$(−13, 5)$ and $(17, 18)$

For the following exercises, construct a table and graph the equation by plotting at least three points.

10.

$y = \frac{1}{2} x + 4$

11.

$4 x - 3 y = 6$

Linear Equations in One Variable

For the following exercises, solve for $x .$

12.

$5 x + 2 = 7 x - 8$

13.

$3 (x + 2) - 10 = x + 4$

14.

$7 x - 3 = 5$

15.

$12 - 5 (x + 1) = 2 x - 5$

16.

$\frac{2 x}{3} - \frac{3}{4} = \frac{x}{6} + \frac{21}{4}$

For the following exercises, solve for $x .$ State all x-values that are excluded from the solution set.

17.

$\frac{x}{x^{2} - 9} + \frac{4}{x + 3} = \frac{3}{x^{2} - 9}$ $x \neq 3, −3$

18.

$\frac{1}{2} + \frac{2}{x} = \frac{3}{4}$

For the following exercises, find the equation of the line using the point-slope formula.

19.

Passes through these two points: $(−2, 1), (4, 2) .$

20.

Passes through the point $(- 3, 4)$ and has a slope of $- \frac{1}{3} .$

21.

Passes through the point $(- 3, 4)$ and is parallel to the graph $y = \frac{2}{3} x + 5.$

22.

Passes through these two points: $(5, 1), (5, 7) .$

Models and Applications

For the following exercises, write and solve an equation to answer each question.

23.

The number of males in the classroom is five more than three times the number of females. If the total number of students is 73, how many of each gender are in the class?

24.

A man has 72 ft. of fencing to put around a rectangular garden. If the length is 3 times the width, find the dimensions of his garden.

25.

A truck rental is $25 plus $.30/mi. Find out how many miles Ken traveled if his bill was $50.20.

Complex Numbers

For the following exercises, use the quadratic equation to solve.

26.

$x^{2} - 5 x + 9 = 0$

27.

$2 x^{2} + 3 x + 7 = 0$

For the following exercises, name the horizontal component and the vertical component.

28.

$4 - 3 i$

29.

$−2 - i$

For the following exercises, perform the operations indicated.

30.

$(9 - i) - (4 - 7 i)$

31.

$(2 + 3 i) - (- 5 - 8 i)$

32.

$2 \sqrt{- 75} + 3 \sqrt{25}$

33.

$\sqrt{- 16} + 4 \sqrt{- 9}$

34.

$- 6 i (i - 5)$

35.

${(3 - 5 i)}^{2}$

36.

$\sqrt{- 4} \cdot \sqrt{- 12}$

37.

$\sqrt{- 2} (\sqrt{- 8} - \sqrt{5})$

38.

$\frac{2}{5 - 3 i}$

39.

$\frac{3 + 7 i}{i}$

Quadratic Equations

For the following exercises, solve the quadratic equation by factoring.

40.

$2 x^{2} - 7 x - 4 = 0$

41.

$3 x^{2} + 18 x + 15 = 0$

42.

$25 x^{2} - 9 = 0$

43.

$7 x^{2} - 9 x = 0$

For the following exercises, solve the quadratic equation by using the square-root property.

44.

$x^{2} = 49$

45.

${(x - 4)}^{2} = 36$

For the following exercises, solve the quadratic equation by completing the square.

46.

$x^{2} + 8 x - 5 = 0$

47.

$4 x^{2} + 2 x - 1 = 0$

For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No real solution.

48.

$2 x^{2} - 5 x + 1 = 0$

49.

$15 x^{2} - x - 2 = 0$

For the following exercises, solve the quadratic equation by the method of your choice.

50.

${(x - 2)}^{2} = 16$

51.

$x^{2} = 10 x + 3$

Other Types of Equations

For the following exercises, solve the equations.

52.

$x^{\frac{3}{2}} = 27$

53.

$x^{\frac{1}{2}} - 4 x^{\frac{1}{4}} = 0$

54.

$4 x^{3} + 8 x^{2} - 9 x - 18 = 0$

55.

$3 x^{5} - 6 x^{3} = 0$

56.

$\sqrt{x + 9} = x - 3$

57.

$\sqrt{3 x + 7} + \sqrt{x + 2} = 1$

58.

$| 3 x - 7 | = 5$

59.

$| 2 x + 3 | - 5 = 9$

Linear Inequalities and Absolute Value Inequalities

For the following exercises, solve the inequality. Write your final answer in interval notation.

60.

$5 x - 8 \leq 12$

61.

$- 2 x + 5 > x - 7$

62.

$\frac{x - 1}{3} + \frac{x + 2}{5} \leq \frac{3}{5}$

63.

$| 3 x + 2 | + 1 \leq 9$

64.

$| 5 x - 1 | > 14$

65.

$| x - 3 | < −4$

For the following exercises, solve the compound inequality. Write your answer in interval notation.

66.

$−4 < 3 x + 2 \leq 18$

67.

$3 y < 1 - 2 y < 5 + y$

For the following exercises, graph as described.

68.

Graph the absolute value function and graph the constant function. Observe the points of intersection and shade the x-axis representing the solution set to the inequality. Show your graph and write your final answer in interval notation.

$| x + 3 | \geq 5$

69.

Graph both straight lines (left-hand side being y1 and right-hand side being y2) on the same axes. Find the point of intersection and solve the inequality by observing where it is true comparing the y-values of the lines. See the interval where the inequality is true.

$x + 3 < 3 x - 4$