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square root of a number : Ifn2=m,n2=m,thennis a square root ofm. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
term : A term is a constant, or the product of a constant and one or more variables. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
variable : A variable is a letter that represents a number whose value may change. | https://openstax.org/books/intermediate-algebra-2e/pages/1-key-terms |
How to determine whether a number is a solution to an equationStep 1.Substitute the number in for the variable in the equation.Step 2.Simplify the expressions on both sides of the equation.Step 3.Determine whether the resulting equation is true.If it is true, the number is a solution.If it is not true, the number is no... | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 1.Substitute the number in for the variable in the equation. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 2.Simplify the expressions on both sides of the equation. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 3.Determine whether the resulting equation is true.If it is true, the number is a solution.If it is not true, the number is not a solution. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
How to Solve Linear Equations Using a General StrategyStep 1.Simplify each side of the equation as much as possible.Use the Distributive Property to remove any parentheses.Combine like terms.Step 2.Collect all the variable terms on one side of the equation.Use the Addition or Subtraction Property of Equality.Step 3.Col... | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 1.Simplify each side of the equation as much as possible.Use the Distributive Property to remove any parentheses.Combine like terms. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 2.Collect all the variable terms on one side of the equation.Use the Addition or Subtraction Property of Equality. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 3.Collect all the constant terms on the other side of the equation.Use the Addition or Subtraction Property of Equality. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 4.Make the coefficient of the variable term equal to 1.Use the Multiplication or Division Property of Equality.State the solution to the equation. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 5.Check the solution.Substitute the solution into the original equation to make sure the result is a true statement. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
How to Solve Equations with Fraction or Decimal CoefficientsStep 1.Find the least common denominator (LCD) ofallthe fractions and decimals (in fraction form) in the equation.Step 2.Multiply both sides of the equation by that LCD. This clears the fractions and decimals.Step 3.Solve using the General Strategy for Solving... | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 1.Find the least common denominator (LCD) ofallthe fractions and decimals (in fraction form) in the equation. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 2.Multiply both sides of the equation by that LCD. This clears the fractions and decimals. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 3.Solve using the General Strategy for Solving Linear Equations. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
How To Use a Problem Solving Strategy for Word ProblemsStep 1.Readthe problem. Make sure all the words and ideas are understood.Step 2.Identifywhat you are looking for.Step 3.Namewhat you are looking for. Choose a variable to represent that quantity.Step 4.Translateinto an equation. It may be helpful to restate the pro... | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 1.Readthe problem. Make sure all the words and ideas are understood. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 2.Identifywhat you are looking for. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 3.Namewhat you are looking for. Choose a variable to represent that quantity. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 4.Translateinto an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 5.Solvethe equation using proper algebra techniques. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 6.Checkthe answer in the problem to make sure it makes sense. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 7.Answerthe question with a complete sentence. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
How To Find Percent ChangeStep 1.Find the amount of changechange=new amountâoriginal amountchange=new amountâoriginal amountStep 2.Find what percent the amount of change is of the original amount.change is what percent of the original amount?change is what percent of the original amount? | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 1.Find the amount of changechange=new amountâoriginal amountchange=new amountâoriginal amount | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 2.Find what percent the amount of change is of the original amount.change is what percent of the original amount?change is what percent of the original amount? | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Discountamount of discount=discount rate·original pricesale price=original amountâdiscountamount of discount=discount rate·original pricesale price=original amountâdiscount | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Mark-upamount of mark-up=mark-up rate·original costlist price=original cost+mark upamount of mark-up=mark-up rate·original costlist price=original cost+mark up | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Simple InterestIf an amount of money,P, called the principal, is invested or borrowed for a period oftyears at an annual interest rater, the amount of interest,I, earned or paid is:I=interestI=PrtwhereP=principalr=ratet=timeI=interestI=PrtwhereP=principalr=ratet=time | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
How To Solve Geometry ApplicationsStep 1.Readthe problem and make sure all the words and ideas are understood.Step 2.Identifywhat you are looking for.Step 3.Namewhat you are looking for by choosing a variable to represent it. Draw the figure and label it with the given information.Step 4.Translateinto an equation by wr... | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 1.Readthe problem and make sure all the words and ideas are understood. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 2.Identifywhat you are looking for. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 3.Namewhat you are looking for by choosing a variable to represent it. Draw the figure and label it with the given information. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 4.Translateinto an equation by writing the appropriate formula or model for the situation. Substitute in the given information. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 5.Solvethe equation using good algebra techniques. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 6.Checkthe answer in the problem and make sure it makes sense. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 7.Answerthe question with a complete sentence. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
The Pythagorean TheoremIn any right triangle, whereaandbare the lengths of the legs, andcis the length of the hypotenuse, the sum of the squares of the lengths of the two legs equals the square of the length of the hypotenuse. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
In any right triangle, whereaandbare the lengths of the legs, andcis the length of the hypotenuse, the sum of the squares of the lengths of the two legs equals the square of the length of the hypotenuse. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Total Value of CoinsFor the same type of coin, the total value of a number of coins is found by using the modelnumber·value=totalvaluenumber·value=totalvaluenumberis the number of coinsvalueis the value of each cointotal valueis the total value of all the coins | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
numberis the number of coins | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
valueis the value of each coin | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
total valueis the total value of all the coins | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
How to solve coin word problems.Step 1.Readthe problem. Make sure all the words and ideas are understood.Determine the types of coins involved.Create a table to organize the information.ââLabel the columns âtype,â ânumber,â âvalue,â âtotal value.âââList the types of coins.ââWrite in the valu... | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 1.Readthe problem. Make sure all the words and ideas are understood.Determine the types of coins involved.Create a table to organize the information.ââLabel the columns âtype,â ânumber,â âvalue,â âtotal value.âââList the types of coins.ââWrite in the value of each type of coin.ââWri... | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 2.Identifywhat you are looking for. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 3.Namewhat you are looking for. Choose a variable to represent that quantity.Use variable expressions to represent the number of each type of coin and write them in the table.Multiply the number times the value to get the total value of each type of coin. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 4.Translateinto an equation.It may be helpful to restate the problem in one sentence with all the important information. Then, translate the sentence into an equation.Write the equation by adding the total values of all the types of coins. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 5.Solvethe equation using good algebra techniques. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 6.Checkthe answer in the problem and make sure it makes sense. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 7.Answerthe question with a complete sentence. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
How To Solve a Uniform Motion ApplicationStep 1.Readthe problem. Make sure all the words and ideas are understood.Draw a diagram to illustrate what it happening.Create a table to organize the information.ââLabel the columns rate, time, distance.ââList the two scenarios.ââWrite in the information you know.St... | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 1.Readthe problem. Make sure all the words and ideas are understood.Draw a diagram to illustrate what it happening.Create a table to organize the information.ââLabel the columns rate, time, distance.ââList the two scenarios.ââWrite in the information you know. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 2.Identifywhat you are looking for. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 3.Namewhat you are looking for. Choose a variable to represent that quantity.Complete the chart.Use variable expressions to represent that quantity in each row.Multiply the rate times the time to get the distance. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 4.Translateinto an equation.Restate the problem in one sentence with all the important information.Then, translate the sentence into an equation. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 5.Solvethe equation using good algebra techniques. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 6.Checkthe answer in the problem and make sure it makes sense. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 7.Answerthe question with a complete sentence. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Inequalities, Number Lines, and Interval Notationx>axâ¥ax<axâ¤ax>axâ¥ax<axâ¤a | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Linear InequalityAlinear inequalityis an inequality in one variable that can be written in one of the following forms wherea,b, andcare real numbers andaâ0:aâ0:ax+b<c,ax+bâ¤c,ax+b>c,ax+bâ¥c.ax+b<c,ax+bâ¤c,ax+b>c,ax+bâ¥c. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Alinear inequalityis an inequality in one variable that can be written in one of the following forms wherea,b, andcare real numbers andaâ0:aâ0:ax+b<c,ax+bâ¤c,ax+b>c,ax+bâ¥c.ax+b<c,ax+bâ¤c,ax+b>c,ax+bâ¥c. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Addition and Subtraction Property of InequalityFor any numbersa,b, andc, ifa<b,thena<b,thena+c<b+caâc<bâca+c>b+caâc>bâca+c<b+caâc<bâca+c>b+caâc>bâcWe can add or subtract the same quantity from both sides of an inequality and still keep the inequality. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
For any numbersa,b, andc, ifa<b,thena<b,thena+c<b+caâc<bâca+c>b+caâc>bâca+c<b+caâc<bâca+c>b+caâc>bâc | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
We can add or subtract the same quantity from both sides of an inequality and still keep the inequality. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Multiplication and Division Property of InequalityFor any numbersa,b, andc,multiply or divide by apositiveifa<bandc>0,thenac<bcandac<bc.ifa>bandc>0,thenac>bcandac>bc.multiply or divide by anegativeifa<bandc<0,thenac>bcandac>bc.ifa>bandc<0,thenac<bcandac<bc.multiply or divide by apositiveifa<bandc>0,thenac<bcandac<bc.if... | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
For any numbersa,b, andc,multiply or divide by apositiveifa<bandc>0,thenac<bcandac<bc.ifa>bandc>0,thenac>bcandac>bc.multiply or divide by anegativeifa<bandc<0,thenac>bcandac>bc.ifa>bandc<0,thenac<bcandac<bc.multiply or divide by apositiveifa<bandc>0,thenac<bcandac<bc.ifa>bandc>0,thenac>bcandac>bc.multiply or divide by ... | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Phrases that indicate inequalities>>â¥â¥<<â¤â¤is greater thanis more thanis larger thanexceedsis greater than or equal tois at leastis no less thanis the minimumis less thanis smaller thanhas fewer thanis lower thanis less than or equal tois at mostis no more thanis the maximum | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
How to solve a compound inequality with âandâStep 1.Solve each inequality.Step 2.Graph each solution. Then graph the numbers that makebothinequalities true. This graph shows the solution to the compound inequality.Step 3.Write the solution in interval notation. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 1.Solve each inequality. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 2.Graph each solution. Then graph the numbers that makebothinequalities true. This graph shows the solution to the compound inequality. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 3.Write the solution in interval notation. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Double InequalityAdouble inequalityis a compound inequality such asa<x<ba<x<b. It is equivalent toa<xa<xandx<b.x<b.Other forms:a<x<bis equivalent toa<xandx<baâ¤xâ¤bis equivalent toaâ¤xandxâ¤ba>x>bis equivalent toa>xandx>baâ¥xâ¥bis equivalent toaâ¥xandxâ¥bOther forms:a<x<bis equivalent toa<xandx<baâ¤xâ¤bis equ... | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Adouble inequalityis a compound inequality such asa<x<ba<x<b. It is equivalent toa<xa<xandx<b.x<b.Other forms:a<x<bis equivalent toa<xandx<baâ¤xâ¤bis equivalent toaâ¤xandxâ¤ba>x>bis equivalent toa>xandx>baâ¥xâ¥bis equivalent toaâ¥xandxâ¥bOther forms:a<x<bis equivalent toa<xandx<baâ¤xâ¤bis equivalent toaâ¤xan... | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
How to solve a compound inequality with âorâStep 1.Solve each inequality.Step 2.Graph each solution. Then graph the numbers that make either inequality true.Step 3.Write the solution in interval notation. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 1.Solve each inequality. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 2.Graph each solution. Then graph the numbers that make either inequality true. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 3.Write the solution in interval notation. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Absolute ValueThe absolute value of a number is its distance from 0 on the number line.The absolute value of a numbernis written as|n||n|and|n|â¥0|n|â¥0for all numbers.Absolute values are always greater than or equal to zero. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Absolute Value EquationsFor any algebraic expression,u, and any positive real number,a,if|u|=athenu=âaoru=aif|u|=athenu=âaoru=aRemember that an absolute value cannot be a negative number. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
How to Solve Absolute Value EquationsStep 1.Isolate the absolute value expression.Step 2.Write the equivalent equations.Step 3.Solve each equation.Step 4.Check each solution. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 1.Isolate the absolute value expression. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 2.Write the equivalent equations. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 3.Solve each equation. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 4.Check each solution. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Equations with Two Absolute ValuesFor any algebraic expressions,uandv,if|u|=|v|thenu=âvoru=vif|u|=|v|thenu=âvoru=v | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Absolute Value Inequalities with<<orâ¤â¤For any algebraic expression,u, and any positive real number,a,if|u|<a,thenâa<u<aif|u|â¤a,thenâaâ¤uâ¤aif|u|<a,thenâa<u<aif|u|â¤a,thenâaâ¤uâ¤a | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
How To Solve Absolute Value Inequalities with<<orâ¤â¤Step 1.Isolate the absolute value expression.Step 2.Write the equivalent compound inequality.|u|<ais equivalent toâa<u<a|u|â¤ais equivalent toâaâ¤uâ¤a|u|<ais equivalent toâa<u<a|u|â¤ais equivalent toâaâ¤uâ¤aStep 3.Solve the compound inequality.Step 4.... | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 1.Isolate the absolute value expression. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 2.Write the equivalent compound inequality.|u|<ais equivalent toâa<u<a|u|â¤ais equivalent toâaâ¤uâ¤a|u|<ais equivalent toâa<u<a|u|â¤ais equivalent toâaâ¤uâ¤a | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 3.Solve the compound inequality. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 4.Graph the solution | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 5.Write the solution using interval notation | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Absolute Value Inequalities with>>orâ¥â¥For any algebraic expression,u, and any positive real number,a,if|u|>a,thenu<âaoru>aif|u|â¥a,thenuâ¤âaoruâ¥aif|u|>a,thenu<âaoru>aif|u|â¥a,thenuâ¤âaoruâ¥a | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
How To Solve Absolute Value Inequalities with>>orâ¥â¥Step 1.Isolate the absolute value expression.Step 2.Write the equivalent compound inequality.|u|>ais equivalent tou<âaoru>a|u|â¥ais equivalent touâ¤âaoruâ¥a|u|>ais equivalent tou<âaoru>a|u|â¥ais equivalent touâ¤âaoruâ¥aStep 3.Solve the compound inequa... | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 1.Isolate the absolute value expression. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 2.Write the equivalent compound inequality.|u|>ais equivalent tou<âaoru>a|u|â¥ais equivalent touâ¤âaoruâ¥a|u|>ais equivalent tou<âaoru>a|u|â¥ais equivalent touâ¤âaoruâ¥a | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
Step 3.Solve the compound inequality. | https://openstax.org/books/intermediate-algebra-2e/pages/2-key-concepts |
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