5.12.5: Chapter Review
Algebraic Expressions
Translate from algebra to words: \(2x + 3\)
Translate from words to algebra: the quotient of \(5x\) and 7.
Translate from an English phrase to an expression: A gym charges $5.00 per class \(c\) and a $20 membership fee.
Use parentheses to make the following statement true: \({2^2} \div + {\rm{ }}5 - 4 \times 3 = - 5\)
Evaluate and simplify \(x^2- 5x\) when \(x = 5\).
Perform the indicated operation for the expression: \((3{x^2} + 2x + 1) + ({x^2} - 2x + 2 - ({x^2} - 3x + 11)\)
Linear Equations in One Variable with Applications
Solve the linear equations using a general strategy: \(3x - 7 = 8\)
Solve the linear equations using properties of equations: \((5x - 3) + 7= 9x - 2(x + 7)\)
It costs 30 cents for an ear of corn. Construct a linear equation and solve how much it costs to buy 23 ears of corn.
State whether the following equation has exactly one solution, no solution, or infinitely many solutions \(5x + 4 = 7x - 14\)
Solve the formula \(A=\frac{\left(b_1+b_2\right)}{2} h\) for \(b_1\)
Linear Inequalities in One Variable with Applications
Graph the inequality \(x > 8\) on a number line and write the interval notation.
Solve the inequality \(- 2x < - 6\), graph the solution on the number line, and write the solution in interval notation.
Construct a linear inequality to solve the application: Daniel wants to surprise his girlfriend with a birthday party at her favorite restaurant. It will cost $42.75 per person for dinner, including tip and tax. His budget for the party is $500. What is the maximum number of people Daniel can have at the party?
Ratio and Proportions
Christer opened a bag of marbles and counted the number of each color. They found they had 9 green, 4 yellow, 13 black, 11 orange, 8 blue, and 7 red.
What is the ratio of green marbles to orange marbles?
What is the ratio of red marbles to blue marbles?
What is the ratio of sum of the marbles with an odd number of marbles to the sum of the marbles with an even number of marbles?
What is the ratio of black marbles to all marbles?
Solve: \(\frac{7}21 = \frac21{x}\)
Basil the cat is 17 pounds and 24 inches long from head to tail. In his new movie Claws , he is supersized to 50 pounds. How many inches long will he be? Round your answer to the nearest tenth.
Graphing Linear Equations and Inequalities
For each ordered pair, decide:
- Is the ordered pair a solution to the equation?
- Is the point on the line in the graph shown?
\({\text{A: }}(0, - 3)\quad \quad {\text{B: }}( - 3,1);\quad \quad {\text{C: }}( - 2, - 4);\quad \quad {\text{D: }}(0,2)\)
Graph \(y = \frac{1}{2}x + 5\) by plotting points.
Determine whether each ordered pair is a solution to the inequality \(y > 2x + 3.\)
\({\text{A: }}(0,0)\quad \quad {\text{B: }}(2,1)\quad \quad {\text{C: }}( - 1, - 5)\quad \quad {\text{D: }}( - 6, - 3)\quad \quad {\text{E: }}(1,0)\)
Write the inequality shown by the graph with the boundary line \(y = 7x - 5\).
Graph the linear inequality: \(y < \frac{1}{3}x + 7\).
Quadratic Equations with Two Variables with Applications
Multiply \((3x - 1)(x + 7)\).
Factor \({x^2} + 8x - 9\).
Graph and list the solutions to the quadratic equation \({x^2} + 11x + 18=0\).
Solve \({x^2} + 7x + 12 = 15x - 3\) by factoring.
Solve \(169{x^2} = 25\) using the square root method.
Solve \({x^2} + 7x + 3 = 0\) using the quadratic formula.
The height in feet, \(h\), of an object shot upwards into the air with initial velocity, \({v_0}\), after \(t\) seconds is given by the formula \(h = - 16{t^2} + {v_0}t\). A firework is shot upwards with initial velocity 130 feet per second. How many seconds will it take to reach a height of 260 feet? Round to the nearest tenth of a second.
Functions
Evaluate the function \(f(x) = {x^2} - 3x + 21\) at the values \(f(-2)\), \(f(-1)\), \(f(0)\), \((1)\), and \(f(2)\).
Determine whether \(\{(-2, 31),\,(-1, 25),\,(1, 19),\,(2, 19)\}\) represents a function.
Determine whether the mappings represent a function:
Determine whether \(5{x^2} + 2{y^2} = 10\) represent \(y\) as a function of \(x\).
Use the vertical line test to determine if the graph represents a function.
Use the set of ordered pairs to find the domain and the range.
\(\{ ( - 3,{\rm{ }}9),{\rm{ }}( - 2,{\rm{ }}4),{\rm{ }}( - 1,{\rm{ }}1),{\rm{ }}(0,{\rm{ }}0),{\rm{ }}(1,{\rm{ }}1),{\rm{ }}(2,{\rm{ }}4),{\rm{ }}(3,{\rm{ }}9)\}\)
Use the graph shown to find the domain and the range.
Graphing Functions
Find the \(x\)- and \(y\)-intercepts on the graph.
Graph \(y = 3x - 3\) using the intercepts.
Find the slope of the line in the graph shown.
Use the slope formula to find the slope of the line between (2, 4) and (5, 7).
Identify the slope and \(y\)-intercept of \(2y = -\frac{2}{3}x + 10\).
Graph the line of \(2y = - \frac{2}{3}x - 4\) using its slope and \(y\)-intercept.
The equation \(P = 28 + 2.54w\) models the relation between the monthly water bill payment, \(P\), in dollars, and the number of units of water, \(w\), used.
Find the payment for a month when 0 units of water were used.
Find the payment for a month when 15 units of water were used.
Interpret the slope and \(P\)-intercept of the equation.
Graph the equation.
System of Linear Equations in Two Variables
Determine if (0, 1) and (2, 3) are solutions to the given system of equations.
\(\left\{ {\begin{array}{*{20}{l}}{x - 2y = 0}\\{3x - y = 5}\end{array}} \right.\)
Solve the system of equations by graphing.
\(\left\{ {\begin{array}{*{20}{l}}{y = 3x + 1}\\{6x - y = 2}\end{array}} \right.\)
Solve the system of equations by substitution.
\(\left\{ {\begin{array}{*{20}{l}}{y = 7x + 2}\\{3x - y = - 6}\end{array}} \right.\)
Solve the systems of equations by elimination.
\(\left\{ {\begin{array}{*{20}{l}}{ - 5x + y = - 15}\\{3x + y = 17}\end{array}} \right.\)
Kenneth currently sells suits for company A at a salary of $22,000 plus a $10 commission for each suit sold. Company B offers him a position with a salary of $28,000 plus a $4 commission for each suit sold. How many suits would Kenneth need to sell for the options to be equal?
Systems of Linear Inequalities in Two Variables
Determine whether (0, 0) and (2, 3) are solutions to the system.
5\\ 2x - y \le 10 \end{array} \right." class=" math-rendered">/**/\left\{ \begin{array}{l} 3x + y > 5\\ 2x - y \le 10 \end{array} \right./**/
Determine whether (0, 0) and (6, –8) are solutions to the darkest shaded region of the graph.
Solve the systems of linear equations by graphing.
\(\left\{ {\begin{array}{*{20}{l}}{y < 3x + 1}\\{6x - y \ge 2}\end{array}} \right.\)
Jocelyn desires to increase both her protein consumption and caloric intake. She desires to have at least 35 more grams of protein each day and no more than an additional 200 calories daily. An ounce of cheddar cheese has 7 grams of protein and 110 calories. An ounce of parmesan cheese has 11 grams of protein and 22 calories.
Write a system of inequalities to model this situation.
Graph the system.
Could she eat 1 ounce of cheddar cheese and 3 ounces of parmesan cheese?
Could she eat 2 ounces of cheddar cheese and 1 ounce of parmesan cheese?
Linear Programming
A toy maker makes two plastic toys, the Ring (\(x \)) and the Stick (\(y \)). The toy maker makes $4 per Ring and $6 per Stick. The Ring uses 3 feet of plastic, while the Stick uses 5 feet of plastic. Today the toy maker has 40 feet of plastic available. The toy maker also only makes 10 plastic toys per day. To maximize profit, how many of each toy should the toy maker make?
Find the objective function.
Write the constraints as a system of inequalities.
Graph of the system of inequalities.
Find the value of the objective function at each corner point of the graphed region.
To maximize his profit, how many of each toy should the toymaker make?