MATH Basic Algebra II  - Unit 9: Mid-Semester Review
Some activities may require making graphs on paper.  Click here to view and print graph paper, as needed.

In the following problems, use the given formulas as needed.

1) Find the slope (m) of the line containing the points (1, 2) and (2, 5). 

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2) State an equation in slope-intercept form of the line containing the points (4, –7) and (9, 3).

Hint:  Calculate the slope using the slope formula, and then find the equation by using the point-slope formula.

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3) State an equation in slope-intercept form that contains the point (–7, 2) and is PARALLEL to the line –x + 3y = 1.

Hint:  Put the equation in slope-intercept form and recall that the slopes of parallel lines are  the same.

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4) Write an equation in slope-intercept form of the line that contains the point (4, 5) and is PERPENDICULAR to the line y = (–1/8)x + 17.
 
Hint:  The slopes of perpendicular lines are opposite reciprocals.

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5) If y varies DIRECTLY as x and y = 32 when x = 8, find the constant of variation, and then write an equation of direct variation that relates the two variables.
 
Hint:  The general direct variation equation is y = kx.

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6) Solve the proportion for x by using cross products.

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7) Solve the proportion for m by using cross products.

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8) Solve the equation for x: 7x – 22 = 3x + 18

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9) A car uses 15 gallons of gasoline to travel 378 miles.  How much gas will the car use to travel 600 miles?  Write a proportion and solve.  Round the answer to the nearest tenth of a gallon.

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10) Two airplanes leave Columbus at the same time and fly in opposite directions.  One plane travels 70 miles per hour faster than the other plane.  After 4 hours they are 3800 miles apart.  What is the rate of each plane?
 
Complete the table, and then write an equation to solve. 
 
Hint:  Distance equals rate times time. (d = r t)

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11) The owner of The Nut House wants to mix 100 pounds of cashews and almonds for a holiday special.  Cashews sell for $8 per pound and almonds sell for $3 per pound.  How many pounds of each type of nut must he mix to obtain a mixture that will sell for $6.25 per pound?
 
Hint:     Let x equal the number of pounds of cashews
 
            Then, 100 – x will equal the number of pounds of almonds.
 
Complete the table.  Find the value of each by multiplying the number of pounds and the price per pound; then, write an equation and solve.

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12) Solve the equation for g.
 
Hint:  Begin by multiplying both sides of the equation by the reciprocal of 1/2.

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13) Solve the inequality for x.

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14) Solve the inequality for x.
 
Recall:  When applying the distributive property, distribute over both terms within parenthesis.

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15)

Solve the absolute value equation for x.   
 
Remember:  Isolate the absolute value part to one side of the equation, and then solve.  There will be two answers.

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16) Solve the absolute value inequality for x. 
 
Hint:  There will be two answers for this absolute value inequality.

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17) Solve the absolute value inequality for x.  
 
Hint:  There will be two answers for this absolute value inequality.

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18) Evaluate using the order of operations.
 
Hint:  Simplify the denominator of the second fraction first.

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19) Evaluate by following the order of operations. 
 
Hint:  Simplify inside the brackets, and then multiply by the 9.

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20) Evaluate using a = –3, b = 2 and c = –1.  Follow the order of operations.
 
Remember:  Multiplication and division take priority over addition; BUT, must be worked left to right as they occur in the problem.

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21) Given the relation: {(–4, 16), (0, 0), (2, 4), (4, 16)},  
          (a)  state the domain, 
          (b)  state the range,  and
          (c)  state if the relation is a function and why or why not.

Remember:  A relation is a function when all x-values are unique.

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22) Simplify.

23) Simplify.
 
Hint:  Write the expression in radical form, then simplify.

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24) Simplify.

25) Simplify by using the properties of exponents.

26) Simplify by using the properties of exponents.
 
Hint:  Any number or expression raised to the zero power equals what?

27) Find f + g.  

28) Find the product of f and g.  

29) Perform the composition for the given functions. 
 
Recall:  When composing functions, substitute the expression of one function in for x in the other function as directed by the given composition.

30) Find the equation of the inverse of the this function:  f (x) = 7x + 3.  State the answer in slope-intercept form.
 
Hint:  Change f (x) to y, then switch x and y, and solve the new equation for y.

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31) Describe the transformation of the given parent function, f (x), to the given transformed function, g (x). Reference the direction of the opening of the function, the horizontal translation, and the vertical translation.
 
Recall:  The general form for an absolute value function is  f (x) = a|x – h| + k.

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32) What are the equations for the piecewise graph? 
 
Hint:  Determine the slope and y-intercept for each linear section of the graph by counting out the slope (rise/run) and locating the y-intercept (0, b).  Then, write each equation in slope-intercept form.  (y = mx + b)

33) Use substitution to solve the system of equations.  Express the answer as an ordered pair.

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34) Use elimination to solve the system of equations.  Express the answer as an ordered pair.

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35) A riverboat travels 36 miles downstream in 2 hours. The return trip takes 3 hours. Write a system of equations and solve.  
          (a)  What is the rate of the riverboat in still water?
          (b)  What is the rate of the current?
 
Hint:  Let r equal rate of the boat in still water.  Let c equal the rate of the current.
 
Organize the given information in the table and then write a system of equations to solve by modeling the equations after the formula, distance equals rate times time. (d = rt).

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36) Tyler and Haley are selling cookie dough for a school fundraiser.  Customers have the choice of chocolate chip cookie dough or sugar cookie dough.  Tyler sold 8 packages of chocolate chip cookie 12 packages of sugar cookie dough for a total of $244.  Haley sold 5 packages of chocolate chip cookie dough and 2 packages of sugar cookies for a total of $92.  Write a system of equations and solve to find the cost of one package of chocolate chip cookie dough and one package of sugar cookie dough.
 
Hint:  Model the equations based on the highlighted information.

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37) The graph of what inequality is shown below?  
 
Hint:  Count out the slope (rise/run), locate the y-intercept (0, b) and test point (0, 0).  The slope-intercept form of an equation is y = mx + b which is helpful in determining the boundary line of the inequality.

38) The graph of what system of inequalities is shown below? 
 
Hint:  Count out the slope (rise/run), locate the y-intercept (0, b) and test point (0, 0) for each inequality.  The slope-intercept form of an equation is y = mx + b which is helpful in determining the boundary line of each inequality.
 
Note:  The dark shaded area (dark brown) represents the solution (intersection points) of the two inequalities.

39) What are coordinates of the vertices of the feasible region formed by the given system?
 
Hint:  Graph these inequalities. Find the vertices of the triangle formed by the overlapping areas.

40) What is the maximum value of the objective function P = 2x –y for the feasible region found in the previous problem?
 
Hint: Substitute each point into the objective function P.  Find the point that gives the maximum value.

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