By LearningExpress Editors

Updated on Aug 24, 2011

To review these concepts, go to Algebra in Word Problems Study Guide.

### Practice

**Directions**: Write each of the following phrases as an algebraic expression.

- eight more than a number
- a number divided by six
- eleven times the sum of a number and five
- the quotient of a number and the difference between nineteen and two
- twice the product of two different numbers
- one more than the difference between six and a number
- the square of a number, minus five times that same number
- ten fewer than fifteen times the cube of a number
- If the quotient of a number and nine is twelve, what is the number?
- If the product of a number and ten is twenty more than fifty, what is the number?
- Eight greater than a number is equal to three times the number. Find the number.
- Six times a number plus three is the same as two fewer than seven times the number. What is the number?
- One-third of a number equals four less than half the number. What is the number?
- Five less than the square of a number is one more than five times the number. Find the two values that could be the number.
- Three times the square of a number is fourteen more than the number. What two values could that number be?
- The difference between a number and two is greater than eight. Find the set of values that describes the number.
- Ten times a number is greater than or equal to three times that number plus fourteen. What is the set of values that describes the number?
- Nine less than a number multiplied by four is less than six times the number minus one. Find the set of values that describes the number.
- Negative eight times a number is greater than or equal to negative five times the number plus eighteen. Describe the set of values that could be the number.
- Three times a number, less than six, is less than the sum of twice the number and eleven. Find the set of values that describes the number.

### Solutions

- Write
*eight*as "8" and a*number*as "*x*": 8 more than*x*. More*than*is a phrase that signals addition, so 8 more than*x*is*x*+ 8. - Write a
*number*as "*x*" and*six*as "6":*x*divided by 6. Divided by is a phrase that signals division, so*x*divided by 6 is - Write
*eleven*as "11," a*number*as "*x*," and*five*as "5": 11 times the sum of*x*and 5. We must find the sum of*x*and 5 first.*Sum*signals addition, so the sum of*x*and 5 is*x*+ 5.*Times*signals multiplication, so 11 times*x*+ 5 is 11(*x*+ 5). We need parentheses so that 11 is multiplied by the sum and not just multiplied by*x*. - Write a
*number*as "*x*,"*nineteen*as "19," and*two*as "2": the quotient of*x*and the difference between 19 and 2. Before we can find the quotient, we must find the difference between 19 and 2.*Difference*signals subtraction, so the difference between 19 and 2 is 19 – 2.*Quotient*signals division, so the quotient of a number and the difference between 19 and 2 is . We need parentheses so that*x*is divided by the difference between 19 and 2 and not just divided by 19. - We can write one unknown number as
*x*and the other unknown number as*y*.*Product*signals multiplication, so the product of two different numbers is*xy*. Twice that product is two multiplied by the product, or 2*xy*. - Write
*one*as "1,"*six*as "6," and a*number*as "*x*."*Difference*signals subtraction, so the difference between 6 and*x*is 6 –*x*. More*than*signals addition, so 1 more than 6 –*x*is (6 –*x*) + 1. The parentheses aren't required; either the subtraction or the addition can be performed first, but it is a good habit to group the operation that is to be performed first in parentheses. - Write a
*number*as "*x*" and*five*as "5": the square of*x*, minus 5 times*x*. The square of*x*is*x*^{2}.*Times*signals multiplication, so 5 times*x*is 5*x*. Minus signals subtraction, so*x*^{2}minus 5*x*is*x*^{2}– 5*x*. - Write
*ten*as "10,"*fifteen*as "15," and a*number*as "*x*": 10 fewer than 15 times the cube of*x*. The cube of*x*is*x*^{3}.*Times*signals multiplication, so 15 times*x*^{3}is 15*x*^{3}. The words*fewer*than signal subtraction and are also a backward phrase: 10 fewer than 15*x*^{3}is 15*x*^{3}– 10. - Write a
*number*as "*x*," write*nine*as "9" and write*twelve*as "12." Replace the word is with the equals sign: the quotient of*x*and 9 = 12.*Quotient*signals division. The quotient of*x*and 9 is . We now have = 12. Solve the equation by multiplying both sides by 9:*x*= 108. - Write a
*number*as "*x*," write*ten*as "10," write*twenty*as "20," and write*fifty*as "50." Replace the word is with the equals sign: the product of*x*and 10 = 20 more than 50.*Product*signals multiplication and*more than*signals addition. The product of*x*and 10 = 20 more than 50 becomes 10*x*= 50 + 20. Solve the equation: 10*x*= 70,*x*= 7. - Write
*eight*as "8," write a*number*as "*x*," and write*three*as "3." Replace*is equal*to with the equals sign: 8 greater than*x*= 3 times*x*.*Greater than*signals addition, and*times*signals multiplication. The number sentence 8 greater than*x*= 3 times*x*becomes*x*+ 8 = 3*x*. Subtract*x*from both sides of the equation:*x*–*x*+ 8 = 3*x*–*x*, 8 = 2*x*. Divide both sides of the equation by 2: ,*x*= 4 - Write
*six*as "6," write a*number*as "*x*," write*two*as "2," and write*seven*as "7." Replace*is the same as*with the equals sign: 6 times*x*plus 3 = 2 fewer than 7 times*x*.*Times*signals multiplication,*plus*signals addition, and*fewer than*signals subtraction. Remember,*fewer than*is a backward phrase: 6 times*x*plus 3 = 2 fewer than 7 times*x*becomes 6*x*+ 3 = 7*x*– 2. Subtract 6*x*from both sides of the equation and add 2 to both sides of the equation: 6*x*+ 3 = 7*x*– 2, 3 =*x*– 2,*x*= 5. - Write
*one-third*as "", write a*number*as "*x*," write*four*as "4," and write "". Replace*equals*with the equals sign:*x*= 4 less than*x*.*Less than*signals subtraction. Remember,*less than*is a backward phrase:*x*= 4 less than*x*becomes*x*=*x*– 4. Convert both fractions to sixths: – 4. Subtract*x*from both sides of the equation: –*x*= –4. Multiply both sides by –6:*x*= 24. - Write
*five*as "5," write a*number*as "*x*," write*one*as "1," and write*five*as "5." Replace is with the equals sign: 5 less than the square of*x*= 1 more than 5 times*x*. The*square*of*x*tells us that we must raise*x*to the second power. The square of*x*is*x*^{2}.*Less than*signals subtraction,*times*signals multiplication, and*more than*signals addition. Remember,*less than*and*more than*are both backward phrases. Our equation is now*x*^{2}– 5 = 5*x*+ 1. Subtract 5*x*and 1 from both sides of the equation:*x*^{2}– 5*x*– 6. Factor the equation and set each factor equal to 0: (*x*– 6)(*x*+ 1),*x*– 6 = 0,*x*= 6,*x*+ 1 = 0,*x*= –1. - Write
*three*as "3," write a*number*as "*x*," and write*fourteen*as "14." Replace is with the equals sign: 3 times the square of*x*= 14 more than*x*. The*square*of*x*tells us that we must raise*x*to the second power. The square of*x*is*x*^{2}.*Times*signals multiplication and*more than*signals addition. Remember,*more than*is a backward phrase: 3 times the square of*x*= 14 more than*x*becomes 3*x*^{2}=*x*+ 14. Subtract*x*and 14 from both sides of the equation: 3*x*^{2}–*x*– 14. Factor the equation and set each factor equal to 0: (3*x*– 7)(*x*+ 2), 3*x*– 7 = 0, 3*x*= 7,*x*= ,*x*+ 2 = 0,*x*= –2. - Write a
*number*as "*x*," write*two*as "2," and write*eight*as "8." Replace*is greater than*with the greater than symbol: the difference between*x*and 2 > 8.*Difference*signals subtraction, so we have*x*– 2 > 8, and*x*> 10. - Write
*ten*as "10," write a*number*as "*x*," write*three*as "3," and write*fourteen*as "14." Replace*is greater than or equal to*with the greater than or equal to symbol: 10 times*x*≥ 3 times*x*plus 1.*Times*signals multiplication and*plus*signals addition, so we now have: 10*x*≥ 3*x*+ 14, 7*x*≥ 14,*x*≥ 2. - Write
*nine*as "9," write a*number*as "*x*," write*four*as "4," write*six*as "6," and write*one*as "1." Replace*is less than*with the less than symbol: 9 less than*x*multiplied by 4 < 6 times*x*minus 1.*Less than*signals subtraction, but it is a backward phrase, so 9 less than*x*is*x*– 9.*Multiplied*signals multiplication and*minus*signals subtraction, so we now have: 4*x*– 9 < 6*x*– 1, –2*x*< 8. Divide by –2 and reverse the inequality symbol:*x*> –4. - Write
*negative eight*as "–8," write a*number*as "*x*," write*negative five*as "–5," and write*eighteen*as "18." Replace*is greater than or equal*with the greater than or equal to symbol: –8 times*x*≥ –5 times*x*plus 18.*Times*signals multiplication and*plus*signals addition, so we now have: –8*x*≥ –5*x*+ 18. Add 5*x*to both sides: –3*x*≥ 18. Divide by –3 and reverse the inequality symbol:*x*≤ –6. - Write
*three*as "3," write a*number*as "*x*," write*six*as "6," write*twice*as 2, and write*eleven*as "11." Replace*is less than*with the less than symbol: 3 times*x*, less than 5 < the sum of 2*x*and 11.*Times*signals multiplication and*sum*signals addition.*Less than*is a backward phrase that signals subtraction, so we now have: 6 – 3*x*< 2*x*+ 11. Add 3*x*to both sides: 6 < 5*x*+ 11. Subtract 11 from both sides and divide by 5: –5 < 5*x*, –1 <*x*.

From Express Review Guides: Math Word Problems. Copyright © 2008 by LearningExpress, LLC. All Rights Reserved.

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