Title: Unit 10 Quiz
Course: Technical Math for Industry

Q: What would you do to isolate the variable in the equation below, using only one step?
- Add 5 to both sides of the equation. (Incorrect. This gives an equivalent equation, , but does not get the variable alone on one side of the equation.)
- Subtract 5 from both sides of the equation. (Correct. This will yield an equivalent equation and isolate the variable: , so .)
- Subtract 12 from both sides of the equation. (Incorrect. This gives an equivalent equation, , but does not get the variable alone on one side of the equation.)
- Add 12 to both sides of the equation. (Incorrect. This gives an equivalent equation, , but does not get the variable alone on one side of the equation.)
A: Subtract 5 from both sides of the equation.

Q: Solve .
A: (a choice with no text)
If right: Correct. Multiply both sides of the equation by 6: or or
If wrong: Incorrect. It seems that you divided by 6 instead of multiplying both sides of the equation by 6. Dividing by 6 will not leave you with x, and you must do the same thing to both sides.
If wrong: Incorrect. You may have divided both sides by 7. That would leave you with . This would not mean that x is equal to .
If wrong: Incorrect. It seems that you added 6 instead of multiplying both sides of the equation by 6. Adding 6 to will not leave you with x, and you must do the same thing to both sides.

Q: Solve for x:
A: (a choice with no text)
If wrong: Incorrect. While trying to get the constant terms together on the right side, you may have subtracted 11 from 19. To combine the constant terms, you would need to add 11 to both sides.
If wrong: Incorrect. While trying to get the variable terms together on the left side, you may have added to . To combine the variable terms, you would need to subtract from both sides.
If right: Correct. Add 11 to both sides: Subtract from both sides: Divide both sides by :
If wrong: Incorrect. You may have made two mistakes. While trying to get the constant terms together on the right side, you may have subtracted 11 from 19. To combine the constant terms, you would need to add 11 to both sides. While trying to get the variable terms together on the left side, you may have added to . To combine the variable terms, you would need to subtract from both sides.

Q: Solve:
A: (a choice with no text)
If wrong: Incorrect. You may have correctly used the distributive property to simplify the equation to , but then tried to add 55 to combine the constant terms. You need to subtract 55 from both sides to combine these terms.
If right: Correct. Multiply both sides by 5 to clear the fractions: Subtract 55 from both sides: Add to both sides: Divide both sides by 4:
If wrong: Incorrect. You may have correctly used the distributive property to simplify the equation to , but then tried to subtract to combine the variable terms. You need to add to both sides to combine these terms.
If wrong: Incorrect. You may have correctly chosen to multiply both sides by 5 to clear the fractions, but forgot to multiply the constant terms, 11 and 3, by 5. Make sure that you use the distributive property when you multiply each side of an equation by a number.

Q: Solve for x.
- There is no solution. (Incorrect. You may have forgotten to distribute the 4 on the left side. This would give you the incorrect equation . The left side should be .)
- all real numbers (Correct. Use the distributive property on the left side to simplify: Combine like terms on both sides: Subtract from both sides: This is true no matter what x is, so the solution is all real numbers.)
A: all real numbers
If wrong: Incorrect. You may have mistakenly combined the and on the left side, as well as the 7 and , before distributing the 4. This is the incorrect order of operations. Multiplication comes before subtraction. The left side should first be simplified to .
If wrong: Incorrect. You may have mistakenly combined the and on the left side, as well as the 7 and , and then multiplied only the variable term by 4. This is the incorrect order of operations. Multiplication comes before addition and subtraction. The left side should first be simplified to .

Q: Alana and Petra go to a used book sale, where every book is the same price. Alana buys 5 books and Petra buys 4 books. They run into their friend Angus, who really loves to read, and discover that he has bought 19 books. Angus received a $10 discount, but he still spent $5 more than Alana and Petra combined. How much does one book cost?
- $2.00 (Incorrect. Let the cost of 1 book. Then the amount spent by Alana, the amount spent by Petra, and the amount spent by Angus. At this point, you may have thought Angus's purchase = Alana's purchase plus 5, plus Petra's purchase plus 5. However, Angus's purchase = Alana's purchase + Petra's purchase plus 5, so .)
- $1.50 (Correct. Let the cost of 1 book. Then the amount spent by Alana, the amount spent by Petra, and the amount spent by Angus. Angus spent $5 more than his two friends, so . Simplify: . Subtract from both sides: . Add 10 to both sides: . Divide both sides by 10: .)
- $19.50 (Incorrect. Let the cost of 1 book. You may have applied the discount to each book that Angus bought, instead of to the whole purchase. That is, you thought he spent dollars, when he actually spent dollars.)
- $0.50 (Incorrect. Let the cost of 1 book. Then the amount spent by Alana, the amount spent by Petra, and the amount spent by Angus. At this point, you may have thought Angus's purchase plus 5 = Alana's purchase + Petra's purchase. However, Angus's purchase = Alana's purchase + Petra's purchase plus 5, so .)
A: $1.50

Q: Find the base of a parallelogram with an area (A) of 72 square inches and height (h) of 4 inches. Use the formula for the area of a parallelogram, .
- 288 inches (Incorrect. You may have incorrectly substituted and ended up multiplying 72 and 4. The correct substitution would be .)
- 18 inches (Correct. Substitute 72 for A and 4 for h: Divide both sides by 4:)
- 68 inches (Incorrect. You may have substituted correctly to get , but then incorrectly subtracted 4. You need to divide both sides by 4 to isolate the variable.)
- inch (Incorrect. You may have incorrectly substituted 72 in place of h and 4 in place of A. The correct substitution would be .)
A: 18 inches

Q: An important formula from chemistry is . Solve the formula for t.
A: (a choice with no text)
If wrong: Incorrect. You correctly divided both sides by n, but you divided one side by r while multiplying the other by r. You need to divide both sides by r.
If right: Correct. Divide both sides by : Simplify to:
If wrong: Incorrect. You correctly divided both sides by r, but you divided one side by n while multiplying the other by n. You need to divide both sides by n.
If wrong: Incorrect. To rewrite any equation, you must follow the same rules that you learned earlier. For example, multiply or divide both sides by the same number. To isolate t in the original equation, you need to divide both sides by .

Q: Which inequality is graphed below?
A: (a choice with no text)
If wrong: Incorrect. You may not have recognized that the end point of the solution is an open circle, so is not included in the inequality. Also, the points on the graph are to the right, so they represent numbers that are greater than .
If wrong: Incorrect. You correctly recognized that the solution does not include , but the points on the graph are to the right, so they represent numbers that are greater than .
If wrong: Incorrect. You may not have recognized that the end point of the solution is an open circle, so is not included in the inequality.
If right: Correct. The end point of the solution is an open circle at , so is not included in the inequality. The points on the graph are to the right, so they represent numbers that are greater than . This is the graph of the inequality .

Q: The solution to which inequality is graphed below?
A: (a choice with no text)
If right: Correct. Add 2 to both sides: , so . The numbers greater than are represented by points to the right of , as shown in the graph.
If wrong: Incorrect. You may have subtracted incorrectly when isolating the variable. The solution to this inequality is , and the graph represents the inequality .
If wrong: Incorrect. If you subtract 4 from both sides, you will find that the solution to this inequality is . However, the given graph represents the inequality .
If wrong: Incorrect. You may have subtracted incorrectly when isolating the variable. The solution to this inequality is , and the graph represents the inequality .

Q: The solution to which inequality is graphed below?
A: (a choice with no text)
If wrong: Incorrect. You may have divided incorrectly when isolating the variable, or misread the graph. If you divide both sides by 6 you get . However, the graph shown represents .
If wrong: Incorrect. You may have misread the graph or incorrectly reversed the inequality sign after division by 3. Reverse the inequality sign only after multiplication or division by a negative number. The solution to this inequality is , while the graph shown represents .
If wrong: Incorrect. You may have forgotten to change the direction of the inequality symbol after you divided both sides by . The solution to this inequality is , while the graph shown represents .
If right: Correct. Divide both sides by 8: , so . Since is not part of the solution, the graph will have an open endpoint at . The numbers greater than are represented by points to the right of , and this is what is shown in the graph.

Q: The solution to which inequality is graphed below?
A: (a choice with no text)
If wrong: Incorrect. The solution of this inequality is , so you may have thought that the graph did not include point . However, the closed circle means that is part of the solution.
If wrong: Incorrect. The solution of this inequality is , and the graph shows , so you may have mistakenly added 4 to both sides of the inequality.
If right: Correct. Add 7 to both sides, and then divide by : or or
If wrong: Incorrect. The solution of this inequality is , and the graph shows , so you may have forgotten to reverse the inequality sign when you divided by .

Q: Solve for z.
A: (a choice with no text)
If wrong: Incorrect. You only multiplied the left side of the inequality to clear the fraction, but you have to multiply both sides by 2.
If right: Correct. Multiply both sides by 2: or Add 2 to both sides: or Divide both sides by and reverse the inequality symbol: or
If wrong: Incorrect. You may have correctly cleared the fraction and isolated the variable term. However, when dividing both sides of the inequality by you forgot to reverse the direction of the inequality symbol.
If wrong: Incorrect. You correctly multiplied both sides of the inequality by 2 to clear the fraction, but then you incorrectly subtracted 2 from both sides. You need to add 2 to both sides.

Q: Solve for b.

or
- or (Incorrect. When isolating the variable in the first inequality, you may have forgotten that a negative divided by a negative is a positive. Also, you tried to subtract 4 from both sides of the second inequality, instead of adding 4.)
- or (Incorrect. When isolating the variable in the first inequality, you may have forgotten that a negative divided by a negative is a positive. Also, you did not reverse the inequality symbol after dividing by a negative number. You also changed the direction in the second inequality when you should not have.)
- The solution is all real numbers. (Correct. To solve the first inequality, add 5 to both sides and then divide by . This will reverse the inequality sign and give you . To solve the second inequality, add 4 to both sides to get . The solution is all values of b greater than 1 or less than or equal to 4. Therefore, b could be any number and the solution is all real numbers.)
- or (Incorrect. You did not reverse the inequality symbol when dividing both sides of the first inequality by . You need to remember to reverse the inequality symbol whenever you multiply or divide both sides of an inequality by a negative number.)
A: The solution is all real numbers.

Q: Solve for t.

and
A: (a choice with no text)
If wrong: Incorrect. You mixed up the directions of the symbols when rewriting the solution. The solution is and . This means that the values of t are greater than and less than 3. The compound inequality given does not say this.
If right: Correct. Solve the first inequality by adding 12 to both sides and then dividing by 3 to get . Solve the second inequality by dividing both sides by to get . The solution and can be rewritten as .
If wrong: Incorrect. You may have forgotten to reverse the inequality symbol when dividing both sides of the second inequality by a negative number. The solution to the first inequality is . The solution to the second inequality is . The solution to the compound inequality is the overlap of these two solutions.
If wrong: Incorrect. You may have tried to subtract 12 from both sides of the first inequality instead of adding 12 to both sides. You also reversed the inequality sign. You should have reversed the inequality sign of the second inequality instead, while solving for t.

Q: Solve for k.
A: (a choice with no text)
If wrong: Incorrect. You may have first added 5 to all parts of the inequality and thought that this eliminated the left side of the inequality. You cannot ignore or remove one part of the inequality, even if it has a 0. Instead, you should add 15 to all parts of the inequality in order to isolate .
If right: Correct. Add 15 to all parts: Simplify to: Divide all parts by , and reverse the inequality signs. so or
If wrong: Incorrect. You forgot to reverse the inequality signs when you divided all parts of the inequality by . Also, this answer makes no sense, because it says that is less than , but more than .
If wrong: Incorrect. You tried to subtract 15 from all parts of the inequality to isolate . Instead, you need to add 15 to all parts as your first step.

Q: For which of the following compound inequalities is there no solution?
- and (Correct. Subtract 12 from both sides of the first inequality to get . Then divide both sides by 3 to get . Divide both sides of the second inequality by and reverse the inequality symbol to get . Since the graphs of and have no overlap, there is no solution.)
- and (Incorrect. The solution to this compound inequality is and . Their graphs intersect for , so the simplest description of the solution is any value greater than or equal to 6.)
- and (Incorrect. The solution to this compound inequality is and . Their graphs intersect for , so any value less than or equal to is a solution.)
- and (Incorrect. You may not have reversed the inequality sign in the second inequality when dividing both sides by . The two inequalities simplify to and . The solution to this compound inequality is all numbers greater than 3.)
A: and

Q: Solve for y:
- or (Incorrect. You may have subtracted 10 from as your first step, and then solved and . This means you are subtracting before finding absolute value, and that is not the correct order of operations. You need to add 10 to both sides first, and then solve and .)
- or (Correct. First add 10 to both sides of the equation, then rewrite as: or Subtract 7 from both sides of both equations: or Divide both sides of both equations by 3: or)
A: or
If wrong: Incorrect. You may have correctly added 10 to both sides to get , but then solved . If , then or . In this problem, X is replaced by . So you should be solving and .
If wrong: Incorrect. You may have correctly added 10 to both sides to get , but then solved only . If , then or . In this problem, X is replaced by . So you should be solving two equations, and , but you forgot one of them.

Q: Solve the inequality:
- or (Incorrect. You solved , instead of . You should first simplify to , and then rewrite without absolute value as .)
A: (a choice with no text)
If wrong: Incorrect. You may have correctly simplified the inequality to , but then only found the solution to . You need to find the solution to the compound inequality .
If wrong: Incorrect. You may have tried to divide both sides of the inequality by 3 while ignoring the term. This would result in the incorrect inequality . Instead, you should subtract 9 from both sides as your first step, and then divide by 3. This will give you . Rewrite this without absolute value and solve the compound inequality.
If right: Correct. Subtract 9 from both sides of the inequality: Divide both sides by 3: Rewrite without absolute value: Subtract 7 from all 3 parts:

Q: Which of the following inequalities has no solution?
A: (a choice with no text)
If wrong: Incorrect. You may have thought that there is no solution because the absolute value expression is being compared to 0. However, any value you enter for s will make the inequality true, so the solution is all real numbers.
If wrong: Incorrect. You may have thought that there is no solution since there is a negative number on the right side of the inequality. Always simplify first. If you add 9 to both sides of the inequality you get . This can be solved by the usual method and does have a solution.
If right: Correct. Add 3 to both sides of the inequality: If is at most , then it must be negative. However, an absolute value can never be a negative number, so there is no solution to the inequality.
If wrong: Incorrect. You may have thought that there is no solution since there is a negative number on the right side of the inequality. Always simplify first. If you divide both sides of the inequality by you get . This can be solved by the usual method and does have a solution.