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Math 1: General Strategies and Basic Equations
General Note
💡 - signifies a strategy.
🛑 - Stop and Practice!
✋ - Take a break!
✩♬ ₊˚.🎧⋆☾⋆⁺₊✧General Strategies✩♬ ₊˚.🎧⋆☾⋆⁺₊✧
Actions
Understand
Solve
Answer
Approaches
Use the Answers
Plug-In Technique
Do the Math - Algebra**
Understand
Identify Important Information and terms
Simplify and re-phrase the problem to make it make sense to you
Connect what you have been asked to what you have been told - Make a connection
Ask yourself: “What formula(s) might apply?”
Annotate the Information needed
Solve
Pick an Approach (Answers, Plug-in, Do the Math)
Write the information from the problem in an organized way by using tables, pictures, or equations.
Look for ways to connect the information provided to the information requested and the answer choices, and then solve.
Digital Testers, use scratch paper and keep it neat and numbered.
Answer
Eliminate unreasonable answer choices
Make sure your solution answers the question.
Digital Testers, pick answers as you go and flag (mark for review) any problem you are not sure about so you can go back to it at the end.
✩♬ ₊˚.🎧⋆☾⋆⁺₊✧Using the Answers✩♬ ₊˚.🎧⋆☾⋆⁺₊✧
Answer choices are provided for most test questions. Strong test-takers use the process of elimination to save time and avoid mistakes.
💡 You can use the answers to:
→ Guide your work and direct you to what you should do with the problem. The type of answers can give you a hint as to where the question is going and what technique to use when solving. For instance, if every answer choice has a √3 as part of the answer, it is likely you’re working with a 30-60-90 triangle.
→ Plug back into the problem to solve. If the algebra is too difficult or you are not sure what to do, plugging the answers into the problem may help. Often, the best way to solve and inequality or absolute value problem is by plugging in the answers to the problem.
→ Eliminate the answer choices. The answers are usually in increasing or decreasing order. If the question asks for the largest value or the smallest value, start with the corresponding answer, either A or D, otherwise start with answer choice C. If the answers are in increasing order and C is too big, then D is also too big. You should move to answer choice B. if B is also too big then A must be the right answer. The great thing about this strategy is that if you apply it properly, you only try a maximum of 2 answer choices to find the right response.
Once you identify an Incorrect response, eliminate it by crossing out the letter and the answer with the cross-out function on the test.
✩♬ ₊˚.🎧⋆☾⋆⁺₊✧Plug-In technique✩♬ ₊˚.🎧⋆☾⋆⁺₊✧
Some problems have more unknown quantities/variables than they have equations. Other problems seem complicated because the problem uses unknowns instead of values. The Plug-In technique allows you to choose your own values for one (or more) of the unknowns in the problem.
💡 You can use the Plug-In Technique when:
→ There are unknown in the answer choice (this is the most common scenario)
→ There are more unknowns than equations
→ There is an undefined quantity that could be assigned a value
Let the problem be your guide. Pick numbers that make sense based on what the problem says.
Choose numbers that are easy to work with.
If a problem asks for the least possible value, make everything else as large as possible
If a problem asks for the greatest possible value, make everything else as small as possible
If the problem involves fractions, try plugging in a common denominator.
The number(s) that you plug-in for the unknown(s) in the problem will give you a certain value for your solution. When you plug that same number into the answer choices, look for the one that gives you the same solution. Make sure you check every answer choice because on a rare occasion, more than one answer choice may work given the number you picked. In that case, try a second set of numbers, and you should be able to eliminate all but one of the remaining answer choices.
✩♬ ₊˚.🎧⋆☾⋆⁺₊✧Fractions✩♬ ₊˚.🎧⋆☾⋆⁺₊✧
Many basic equation problems will require you to manipulate fractions. The two most common fraction scenarios involve cross-multiplying and/or finding a least common denominator.
Cross Multiplication
a/b = c/d → ad = cb
You can also flip the fractions if needed:
a/b = c/d → b/a = d/c
Least common denominator
In order to find the least common denominator, look for the smallest multiple that the denominators have in common.
ab/cd = ef/c
Multiply both sides by the least common denominator, cd.
cd(ab/cd) = cd(ef/c) → ab = def
💡 When working with algebraic fractions, eliminate the fractions by:
Cross multiplying when you have fraction = fraction
Multiplying each term by the least common denominator.
✩♬ ₊˚.🎧⋆☾⋆⁺₊✧Solving Partially✩♬ ₊˚.🎧⋆☾⋆⁺₊✧
💡 When you are asked to solve for the value of an expression rather than the value of the individual unknowns, try to find a direct link between the information given and the information requested. Save time by not solving the equation(s) completely.
✋!
✩♬ ₊˚.🎧⋆☾⋆⁺₊Interpreting Equations✩♬ ₊˚.🎧⋆☾⋆⁺₊
There are two main ways that you will be asked to interpret single-variable equations. The first way is by asking you to recognize how many solutions an equation has. The second way is by asking you to interpret the meaning of a term in an equation given its context.
💡 When facing a problem that asks about the number of solutions in a single variable equation, remember how to find the answer in each of the scenarios below.
1 Solution
The coefficient of x and the constant are both different from their counterparts on the other side of the equation.
When you solve the equation, you get exactly one answer.
Example: 3x + 5 = 2x + 9
x = 4
Infinite number of Solutions
The coefficients of x and the constant are both the same as their counterparts on the other side of the equation.
When you solve the equation, you get a statement that is true for all values x.
Example: 82x + 3 = 82x + 3
3 = 3 or 82x = 82x or 0 = 0
No solution
The coefficient of x is the same on both sides of the equation, but the constant is different.
When you solve the equation, you get an untrue statement.
Example: 3x - 2 = 3x + 5
-2 ≠ 5
✩♬ ₊˚.🎧⋆☾⋆⁺₊✧Word Problems✩♬ ₊˚.🎧⋆☾⋆⁺₊✧
You will come across word problems on the PSAT and SAT tests, in which no equation will be given. The trick to answering word problems correctly is to understand how to form the words in the problem into a solvable equation. Knowing the meaning of keywords is particularly useful in interpreting the word problem.
💡 When solving a word problem, read the question throroughly and look for keywords:
Keywords Meaning:
Equals, is, are, was =
Sum, added, more than +
Difference, subtracted, less than -
Times, product, multiple, of x
Divided, per ÷
✩♬ ₊˚.🎧⋆☾⋆⁺₊✧Equivalent Equations✩♬ ₊˚.🎧⋆☾⋆⁺₊✧
💡 If you are given a word problem that is long and wordy, then skip to the question and see what they are asking. If they are asking you to solve for one unknown in terms of the others, then don’t read the problem—just solve for the requested unknown.
✩♬ ₊˚.🎧⋆☾⋆⁺₊✧Case Questions✩♬ ₊˚.🎧⋆☾⋆⁺₊✧
A case question presents you with more than one answer possibility. You must determine which case or combination of cases answers the question.
💡 On a Case Question, it is often possible to eliminate more than one answer at a time. If the case works, then eliminate all answers that DO NOT include that case. If the case does not work, then eliminate all answers that DO include that case.
STEP 1: If applicable, start by testing a case that appears in exactly two answer choices. If it works, eliminate the other two answer choices. If it does not work, eliminate the two choices in which it appears. This guarantees you will have only two choices left to analyze.
STEP 2: Examine the remaining answers and choose another case to test. Sometimes you do not have to test all three cases.
Note: These probably will not appear on the Digital SAT, in other words, a very small chance
✋!
✩♬ ₊˚.🎧⋆☾⋆Free Response Questions✩♬ ₊˚.🎧⋆☾⋆
Free-response questions appear on the digital SAT, but not on the ACT. These problems do not provide any answer choices, but rather require students to type in their own answers.
Remember, for all free response questions:
You cannot enter a mixed number. Convert it to a decimal or improper fraction.
When entering an extended decimal, fill in as many digits as will fit. For any other type of number, the correct answer will fit in the blank.
Some questions may have multiple correct answers. You only need to fill in 1 answer.
The correct answer may be a positive or negative number. The answer can take up a maximum of 5 spaces, not including the possible negative sign. Make sure your answer is typed correctly because the test will count it wrong if you include an extra slash (/) or a decimal (.). No commas are allowed.
💡 → If the question contains the word “possible,” there is more than one solution; solve for only one answer.
→ Think about the concept that is being tested and what restrictions might exist given that concept. If you need to guess, make sure to guess a value that is within the restricted limit (sin and cos are between 0 and 1, an angle in a triangle is between 0 and 180, the probability that an event occurs is between 0 and 1, etc.)
→ The correct answers tend to be positive, whole numbers, or simple fractions. For instance, 2 is more likely to be right than 2.46.
Strats at a Glance:
💡 Use these actions and approaches when solving math problems:
Actions
Understand
Solve
Answer
Approaches
Use the Answers
Plug-In Technique
Do the Math - Algebra**
💡 You can use the answers to:
→ Guide your work and direct you to what you should do with the problem. The type of answers can give you a hint as to where the question is going and what technique to use when solving. For instance, if every answer choice has a √3 as part of the answer, it is likely you’re working with a 30-60-90 triangle.
→ Plug back into the problem to solve. If the algebra is too difficult or you are not sure what to do, plugging the answers into the problem may help. Often, the best way to solve and inequality or absolute value problem is by plugging in the answers to the problem.
→ Eliminate the answer choices. The answers are usually in increasing or decreasing order. If the question asks for the largest value or the smallest value, start with the corresponding answer, either A or D, otherwise start with answer choice C. If the answers are in increasing order and C is too big, then D is also too big. You should move to answer choice B. if B is also too big then A must be the right answer. The great thing about this strategy is that if you apply it properly, you only try a maximum of 2 answer choices to find the right response.
💡 You can use the Plug-In Technique when:
→ There are unknown in the answer choice (this is the most common scenario)
→ There are more unknowns than equations
→ There is an undefined quantity that could be assigned a value
💡 When working with algebraic fractions, eliminate the fractions by:
Cross multiplying when you have fraction = fraction
Multiplying each term by the least common denominator.
💡 When you are asked to solve for the value of an expression rather than the value of the individual unknowns, try to find a direct link between the information given and the information requested. Save time by not solving the equation(s) completely.
💡 When facing a problem that asks about the number of solutions in a single variable equation, remember how to find the answer in each of the scenarios above in the section.
💡 When solving a word problem, read the question throroughly and look for keywords:
Keywords Meaning:
Equals, is, are, was =
Sum, added, more than +
Difference, subtracted, less than -
Times, product, multiple, of x
Divided, per ÷
💡 If you are given a word problem that is long and wordy, then skip to the question and see what they are asking. If they are asking you to solve for one unknown in terms of the others, then don’t read the problem—just solve for the requested unknown.
💡 On a Case Question, it is often possible to eliminate more than one answer at a time. If the case works, then eliminate all answers that DO NOT include that case. If the case does not work, then eliminate all answers that DO include that case.
💡 → If the question contains the word “possible,” there is more than one solution; solve for only one answer.
→ Think about the concept that is being tested and what restrictions might exist given that concept. If you need to guess, make sure to guess a value that is within the restricted limit (sin and cos are between 0 and 1, an angle in a triangle is between 0 and 180, the probability that an event occurs is between 0 and 1, etc.) → The correct answers tend to be positive, whole numbers, or simple fractions. For instance, 2 is more likely to be right than 2.46.
Math 1: General Strategies and Basic Equations Completed!
█▒▒▒▒▒▒▒▒▒10%
████▒▒▒▒▒▒30%
█████▒▒▒▒▒50%
████████▒▒80%
██████████100%
⋘ 𝑙𝑜𝑎𝑑𝑖𝑛𝑔 𝑑𝑎𝑡𝑎... ⋙
⋘ 𝑃𝑙𝑒𝑎𝑠𝑒 𝑤𝑎𝑖𝑡... ⋙
⋘ ᴛʀʏ ʟᴀᴛᴇʀ... ⋙
𝐍𝐨𝐰 𝐥𝐨𝐚𝐝𝐢𝐧𝐠. . .
↷✦; w e l c o m e ❞
𝗡𝗼𝘄 𝗽𝗹𝗮𝘆𝗶𝗻𝗴:
"Math 2: Linear Equations and Algebra"
01:23 ━━━━●───── 03:43
ㅤ ⇄ ◃◃ ⅠⅠ ▹▹ ↻
---˖⁺. ༶ ⋆˙⊹❀♡❀˖⁺. ༶ ⋆˙⊹---
Math 1: General Strategies and Basic Equations
General Note
💡 - signifies a strategy.
🛑 - Stop and Practice!
✋ - Take a break!
✩♬ ₊˚.🎧⋆☾⋆⁺₊✧General Strategies✩♬ ₊˚.🎧⋆☾⋆⁺₊✧
Actions
Understand
Solve
Answer
Approaches
Use the Answers
Plug-In Technique
Do the Math - Algebra**
Understand
Identify Important Information and terms
Simplify and re-phrase the problem to make it make sense to you
Connect what you have been asked to what you have been told - Make a connection
Ask yourself: “What formula(s) might apply?”
Annotate the Information needed
Solve
Pick an Approach (Answers, Plug-in, Do the Math)
Write the information from the problem in an organized way by using tables, pictures, or equations.
Look for ways to connect the information provided to the information requested and the answer choices, and then solve.
Digital Testers, use scratch paper and keep it neat and numbered.
Answer
Eliminate unreasonable answer choices
Make sure your solution answers the question.
Digital Testers, pick answers as you go and flag (mark for review) any problem you are not sure about so you can go back to it at the end.
✩♬ ₊˚.🎧⋆☾⋆⁺₊✧Using the Answers✩♬ ₊˚.🎧⋆☾⋆⁺₊✧
Answer choices are provided for most test questions. Strong test-takers use the process of elimination to save time and avoid mistakes.
💡 You can use the answers to:
→ Guide your work and direct you to what you should do with the problem. The type of answers can give you a hint as to where the question is going and what technique to use when solving. For instance, if every answer choice has a √3 as part of the answer, it is likely you’re working with a 30-60-90 triangle.
→ Plug back into the problem to solve. If the algebra is too difficult or you are not sure what to do, plugging the answers into the problem may help. Often, the best way to solve and inequality or absolute value problem is by plugging in the answers to the problem.
→ Eliminate the answer choices. The answers are usually in increasing or decreasing order. If the question asks for the largest value or the smallest value, start with the corresponding answer, either A or D, otherwise start with answer choice C. If the answers are in increasing order and C is too big, then D is also too big. You should move to answer choice B. if B is also too big then A must be the right answer. The great thing about this strategy is that if you apply it properly, you only try a maximum of 2 answer choices to find the right response.
Once you identify an Incorrect response, eliminate it by crossing out the letter and the answer with the cross-out function on the test.
✩♬ ₊˚.🎧⋆☾⋆⁺₊✧Plug-In technique✩♬ ₊˚.🎧⋆☾⋆⁺₊✧
Some problems have more unknown quantities/variables than they have equations. Other problems seem complicated because the problem uses unknowns instead of values. The Plug-In technique allows you to choose your own values for one (or more) of the unknowns in the problem.
💡 You can use the Plug-In Technique when:
→ There are unknown in the answer choice (this is the most common scenario)
→ There are more unknowns than equations
→ There is an undefined quantity that could be assigned a value
Let the problem be your guide. Pick numbers that make sense based on what the problem says.
Choose numbers that are easy to work with.
If a problem asks for the least possible value, make everything else as large as possible
If a problem asks for the greatest possible value, make everything else as small as possible
If the problem involves fractions, try plugging in a common denominator.
The number(s) that you plug-in for the unknown(s) in the problem will give you a certain value for your solution. When you plug that same number into the answer choices, look for the one that gives you the same solution. Make sure you check every answer choice because on a rare occasion, more than one answer choice may work given the number you picked. In that case, try a second set of numbers, and you should be able to eliminate all but one of the remaining answer choices.
✩♬ ₊˚.🎧⋆☾⋆⁺₊✧Fractions✩♬ ₊˚.🎧⋆☾⋆⁺₊✧
Many basic equation problems will require you to manipulate fractions. The two most common fraction scenarios involve cross-multiplying and/or finding a least common denominator.
Cross Multiplication
a/b = c/d → ad = cb
You can also flip the fractions if needed:
a/b = c/d → b/a = d/c
Least common denominator
In order to find the least common denominator, look for the smallest multiple that the denominators have in common.
ab/cd = ef/c
Multiply both sides by the least common denominator, cd.
cd(ab/cd) = cd(ef/c) → ab = def
💡 When working with algebraic fractions, eliminate the fractions by:
Cross multiplying when you have fraction = fraction
Multiplying each term by the least common denominator.
✩♬ ₊˚.🎧⋆☾⋆⁺₊✧Solving Partially✩♬ ₊˚.🎧⋆☾⋆⁺₊✧
💡 When you are asked to solve for the value of an expression rather than the value of the individual unknowns, try to find a direct link between the information given and the information requested. Save time by not solving the equation(s) completely.
✋!
✩♬ ₊˚.🎧⋆☾⋆⁺₊Interpreting Equations✩♬ ₊˚.🎧⋆☾⋆⁺₊
There are two main ways that you will be asked to interpret single-variable equations. The first way is by asking you to recognize how many solutions an equation has. The second way is by asking you to interpret the meaning of a term in an equation given its context.
💡 When facing a problem that asks about the number of solutions in a single variable equation, remember how to find the answer in each of the scenarios below.
1 Solution
The coefficient of x and the constant are both different from their counterparts on the other side of the equation.
When you solve the equation, you get exactly one answer.
Example: 3x + 5 = 2x + 9
x = 4
Infinite number of Solutions
The coefficients of x and the constant are both the same as their counterparts on the other side of the equation.
When you solve the equation, you get a statement that is true for all values x.
Example: 82x + 3 = 82x + 3
3 = 3 or 82x = 82x or 0 = 0
No solution
The coefficient of x is the same on both sides of the equation, but the constant is different.
When you solve the equation, you get an untrue statement.
Example: 3x - 2 = 3x + 5
-2 ≠ 5
✩♬ ₊˚.🎧⋆☾⋆⁺₊✧Word Problems✩♬ ₊˚.🎧⋆☾⋆⁺₊✧
You will come across word problems on the PSAT and SAT tests, in which no equation will be given. The trick to answering word problems correctly is to understand how to form the words in the problem into a solvable equation. Knowing the meaning of keywords is particularly useful in interpreting the word problem.
💡 When solving a word problem, read the question throroughly and look for keywords:
Keywords Meaning:
Equals, is, are, was =
Sum, added, more than +
Difference, subtracted, less than -
Times, product, multiple, of x
Divided, per ÷
✩♬ ₊˚.🎧⋆☾⋆⁺₊✧Equivalent Equations✩♬ ₊˚.🎧⋆☾⋆⁺₊✧
💡 If you are given a word problem that is long and wordy, then skip to the question and see what they are asking. If they are asking you to solve for one unknown in terms of the others, then don’t read the problem—just solve for the requested unknown.
✩♬ ₊˚.🎧⋆☾⋆⁺₊✧Case Questions✩♬ ₊˚.🎧⋆☾⋆⁺₊✧
A case question presents you with more than one answer possibility. You must determine which case or combination of cases answers the question.
💡 On a Case Question, it is often possible to eliminate more than one answer at a time. If the case works, then eliminate all answers that DO NOT include that case. If the case does not work, then eliminate all answers that DO include that case.
STEP 1: If applicable, start by testing a case that appears in exactly two answer choices. If it works, eliminate the other two answer choices. If it does not work, eliminate the two choices in which it appears. This guarantees you will have only two choices left to analyze.
STEP 2: Examine the remaining answers and choose another case to test. Sometimes you do not have to test all three cases.
Note: These probably will not appear on the Digital SAT, in other words, a very small chance
✋!
✩♬ ₊˚.🎧⋆☾⋆Free Response Questions✩♬ ₊˚.🎧⋆☾⋆
Free-response questions appear on the digital SAT, but not on the ACT. These problems do not provide any answer choices, but rather require students to type in their own answers.
Remember, for all free response questions:
You cannot enter a mixed number. Convert it to a decimal or improper fraction.
When entering an extended decimal, fill in as many digits as will fit. For any other type of number, the correct answer will fit in the blank.
Some questions may have multiple correct answers. You only need to fill in 1 answer.
The correct answer may be a positive or negative number. The answer can take up a maximum of 5 spaces, not including the possible negative sign. Make sure your answer is typed correctly because the test will count it wrong if you include an extra slash (/) or a decimal (.). No commas are allowed.
💡 → If the question contains the word “possible,” there is more than one solution; solve for only one answer.
→ Think about the concept that is being tested and what restrictions might exist given that concept. If you need to guess, make sure to guess a value that is within the restricted limit (sin and cos are between 0 and 1, an angle in a triangle is between 0 and 180, the probability that an event occurs is between 0 and 1, etc.)
→ The correct answers tend to be positive, whole numbers, or simple fractions. For instance, 2 is more likely to be right than 2.46.
Strats at a Glance:
💡 Use these actions and approaches when solving math problems:
Actions
Understand
Solve
Answer
Approaches
Use the Answers
Plug-In Technique
Do the Math - Algebra**
💡 You can use the answers to:
→ Guide your work and direct you to what you should do with the problem. The type of answers can give you a hint as to where the question is going and what technique to use when solving. For instance, if every answer choice has a √3 as part of the answer, it is likely you’re working with a 30-60-90 triangle.
→ Plug back into the problem to solve. If the algebra is too difficult or you are not sure what to do, plugging the answers into the problem may help. Often, the best way to solve and inequality or absolute value problem is by plugging in the answers to the problem.
→ Eliminate the answer choices. The answers are usually in increasing or decreasing order. If the question asks for the largest value or the smallest value, start with the corresponding answer, either A or D, otherwise start with answer choice C. If the answers are in increasing order and C is too big, then D is also too big. You should move to answer choice B. if B is also too big then A must be the right answer. The great thing about this strategy is that if you apply it properly, you only try a maximum of 2 answer choices to find the right response.
💡 You can use the Plug-In Technique when:
→ There are unknown in the answer choice (this is the most common scenario)
→ There are more unknowns than equations
→ There is an undefined quantity that could be assigned a value
💡 When working with algebraic fractions, eliminate the fractions by:
Cross multiplying when you have fraction = fraction
Multiplying each term by the least common denominator.
💡 When you are asked to solve for the value of an expression rather than the value of the individual unknowns, try to find a direct link between the information given and the information requested. Save time by not solving the equation(s) completely.
💡 When facing a problem that asks about the number of solutions in a single variable equation, remember how to find the answer in each of the scenarios above in the section.
💡 When solving a word problem, read the question throroughly and look for keywords:
Keywords Meaning:
Equals, is, are, was =
Sum, added, more than +
Difference, subtracted, less than -
Times, product, multiple, of x
Divided, per ÷
💡 If you are given a word problem that is long and wordy, then skip to the question and see what they are asking. If they are asking you to solve for one unknown in terms of the others, then don’t read the problem—just solve for the requested unknown.
💡 On a Case Question, it is often possible to eliminate more than one answer at a time. If the case works, then eliminate all answers that DO NOT include that case. If the case does not work, then eliminate all answers that DO include that case.
💡 → If the question contains the word “possible,” there is more than one solution; solve for only one answer.
→ Think about the concept that is being tested and what restrictions might exist given that concept. If you need to guess, make sure to guess a value that is within the restricted limit (sin and cos are between 0 and 1, an angle in a triangle is between 0 and 180, the probability that an event occurs is between 0 and 1, etc.) → The correct answers tend to be positive, whole numbers, or simple fractions. For instance, 2 is more likely to be right than 2.46.
Math 1: General Strategies and Basic Equations Completed!
█▒▒▒▒▒▒▒▒▒10%
████▒▒▒▒▒▒30%
█████▒▒▒▒▒50%
████████▒▒80%
██████████100%
⋘ 𝑙𝑜𝑎𝑑𝑖𝑛𝑔 𝑑𝑎𝑡𝑎... ⋙
⋘ 𝑃𝑙𝑒𝑎𝑠𝑒 𝑤𝑎𝑖𝑡... ⋙
⋘ ᴛʀʏ ʟᴀᴛᴇʀ... ⋙
𝐍𝐨𝐰 𝐥𝐨𝐚𝐝𝐢𝐧𝐠. . .
↷✦; w e l c o m e ❞
𝗡𝗼𝘄 𝗽𝗹𝗮𝘆𝗶𝗻𝗴:
"Math 2: Linear Equations and Algebra"
01:23 ━━━━●───── 03:43
ㅤ ⇄ ◃◃ ⅠⅠ ▹▹ ↻
---˖⁺. ༶ ⋆˙⊹❀♡❀˖⁺. ༶ ⋆˙⊹---
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