Unit 14 · Expressions, Equations and Inequalities

Maya’s phone line costs $20 a month plus $5 for every gigabyte she uses. The bill depends on one number that changes each month: the gigabytes. Write that number as the letter g. Then the bill is 20 + 5g. A variable is a letter that stands for a number. Here g is the variable. When g is 3, the bill is 20 + 5 × 3 = 20 + 15 = 35 dollars. When g is 6, the bill is 20 + 30 = 50 dollars.

A variable does two different jobs. Sometimes it is one unknown number waiting to be found: in 7 + n = 12, the letter n can only be 5. Sometimes it is a quantity that varies, taking many values: in 20 + 5g, the letter g can be 0, 1, 2, or any number of gigabytes. Both jobs use the same idea. The letter holds the place of a number so you can do arithmetic before you know the number.

The whole thing, 20 + 5g, is an expression: numbers, letters and operation signs put together with no equals sign. Each part that is added or subtracted is a term, so 20 + 5g has two terms. The 20 is a constant because it never changes. In the term 5g, the 5 is the coefficient, the number multiplied by the variable. Writing 5g means 5 × g. Algebra drops the times sign so it is not confused with the letter x.

Common mistake: reading 5g as 5 + g, or as a two-digit number like 53 when g is 3. Neither is right. 5g with g = 3 is 5 × 3 = 15. A variable written alone, like g, has a coefficient of 1, because g means 1 × g.

A word problem hides an expression inside its sentences. Some phrases point to one operation. Sum, more than, increased by and total mean add. Difference, less than, fewer than and decreased by mean subtract. Product, times, twice and of mean multiply. Quotient, divided by, split into and per mean divide. "The sum of a number and 9" is n + 9. "The product of 6 and a number" is 6n.

Order matters with subtraction and division. "A number decreased by 4" is n − 4. "4 less than a number" is also n − 4, because the 4 is taken away from the number. Students often write 4 − n for "4 less than a number," which says the opposite: it takes the number away from 4. Test with a real value. If the number is 10, then 4 less than it is 6. Now 10 − 4 = 6 works, while 4 − 10 = −6 does not.

Longer phrases build in layers. Take "3 more than twice a number." Twice a number is 2n. Three more than that is 2n + 3. Take "the quotient of a number and 5, decreased by 1." The quotient is n/5. Decreased by 1 gives n/5 − 1. Try "the product of 7 and the sum of a number and 2." The sum is n + 2. The product with 7 needs parentheses: 7(n + 2). Without them, 7n + 2 would multiply only the n.

Many prices have two parts: a fixed amount you pay no matter what, and a rate you pay per item, per mile or per hour. A ride service in Chicago might charge a $3 pickup fee plus $2 per mile (an example, not a real price list). For m miles the cost is 3 + 2m. The 3 is the fixed cost, the constant. The 2 is the rate, the coefficient, because it is paid per mile.

Set up a few more. A pizza shop charges $14 per pizza plus a $4 delivery fee: for p pizzas, 14p + 4. A movie costs $11 per ticket and the group shares one $6 popcorn: for t tickets, 11t + 6. A babysitter earns $12 per hour with no fixed fee: for h hours, 12h. When there is no fixed amount, the expression has no constant.

The most common mistake is swapping the two numbers. For the pizza shop, 4p + 14 would mean $4 per pizza and a $14 fee, which is a different shop. To check, ask: which number gets bigger when I buy more? That number multiplies the variable. Test with p = 2: 14 × 2 + 4 = 32 dollars for two pizzas delivered, which makes sense. The swapped version gives 4 × 2 + 14 = 22, too cheap for two pizzas.

Two students evaluate 3 + 4 × 2. One gets 14 by adding first. The other gets 11 by multiplying first. Only one can be right, so mathematicians agreed on an order of operations. First, do anything inside grouping symbols such as parentheses. Second, do any exponent. Third, multiply and divide, working left to right. Last, add and subtract, working left to right. Under this rule, 3 + 4 × 2 = 3 + 8 = 11.

Work through 20 − 2 × 3^2. The exponent comes first: 3^2 = 3 × 3 = 9. Then multiply: 2 × 9 = 18. Then subtract: 20 − 18 = 2. A student who multiplies 2 × 3 first gets 6^2 = 36, then 20 − 36 = −16, which is wrong. Now try (8 + 4) ÷ 2 × 3. Parentheses first: 12 ÷ 2 × 3. Division and multiplication have the same rank, so go left to right: 12 ÷ 2 = 6, then 6 × 3 = 18. Doing 2 × 3 first gives 12 ÷ 6 = 2, which is wrong.

A fraction bar works like parentheses around the top and the bottom. In (10 + 6)/4, add first to get 16/4 = 4. Subtraction also goes left to right: 10 − 4 − 3 means (10 − 4) − 3 = 3, not 10 − 1 = 9. When in doubt, write each step on its own line and change only one thing per line.

To evaluate an expression means to find its value for given numbers. First substitute: replace each variable with its number, using parentheses around the number. Then compute with the order of operations. Evaluate 20 + 5g for g = 7: write 20 + 5(7), multiply to get 20 + 35, then add to get 55. Maya’s bill for 7 gigabytes is $55.

Expressions can have more than one variable. Evaluate 3a − 2b for a = 6 and b = 4: write 3(6) − 2(4) = 18 − 8 = 10. Evaluate x^2 + 4x for x = 3: write 3^2 + 4(3) = 9 + 12 = 21. Notice that the parentheses keep 4(3) from turning into the number 43. Fractions and decimals work the same way: for h = 2.5, the babysitter’s pay 12h is 12 × 2.5 = 30 dollars.

Watch for two mistakes. First, an exponent applies only to what is right next to it. For x = 5, 2x^2 means 2 × 25 = 50, not (2 × 5)^2 = 100. Second, substituting a negative number needs parentheses. For y = −3, y^2 means (−3)^2 = 9, because a negative times a negative is positive. Writing −3^2 without parentheses is read as −(3^2) = −9, a different value.

An expression with a variable that varies can be turned into a table of values. Choose several values for the variable, evaluate for each one, and write the pairs in rows. For 20 + 5g with g = 0, 1, 2, 3, the values are 20, 25, 30, 35. Each time the input g goes up by 1, the value goes up by 5, the coefficient. The starting value when g = 0 is 20, the constant.

Tables reveal patterns that a single number cannot. Compare 3 + 2m (a ride with a $3 fee and $2 per mile) with 5m (a rival with $5 per mile and no fee). At m = 1: 5 and 5. At m = 2: 7 and 10. At m = 3: 9 and 15. The first plan wins for any trip longer than 1 mile, because it grows by 2 each mile while the second grows by 5.

Read a table backward, too. If a row shows m = 4 and cost 11, check that 3 + 2(4) = 3 + 8 = 11. If someone claims a row with m = 6 and cost 14, test it. Since 3 + 2(6) = 15, the row is wrong. A common error is starting the pattern in the wrong place. A student might write 3, 5, 7, 9 for m = 1, 2, 3, 4. The correct values are 5, 7, 9, 11, because the constant 3 belongs to m = 0.

Suppose 4 friends each buy a $7 ticket and a $2 drink. You can add first and then multiply: 4 × (7 + 2) = 4 × 9 = 36. Or you can multiply each part and then add: 4 × 7 + 4 × 2 = 28 + 8 = 36. Same answer. This is the distributive property: a(b + c) = ab + ac. The multiplier is handed out, or distributed, to each term inside the parentheses.

With a variable, the property lets you rewrite an expression in a new form with the same value. To expand 3(x + 5), multiply 3 by x and 3 by 5 to get 3x + 15. Expand 2(4n − 1): 2 × 4n = 8n and 2 × 1 = 2, so 8n − 2. Expand 5(2a + 3b): 10a + 15b. Two expressions that have the same value for every value of the variable are equivalent expressions. So 3(x + 5) and 3x + 15 are equivalent. Check with x = 2: 3(7) = 21 and 6 + 15 = 21.

The classic mistake is to multiply only the first term, writing 3(x + 5) as 3x + 5. Test it with x = 1: 3(6) = 18, but 3 + 5 = 8, so the two are not equal. Another slip is with subtraction: 6(y − 4) is 6y − 24, not 6y − 4. Draw an arrow from the multiplier to each term until distributing feels automatic.

The property also makes mental math easier. To find 7 × 98, think 7 × (100 − 2) = 700 − 14 = 686. To find 12 × 15, think 12 × (10 + 5) = 120 + 60 = 180. Check: 12 × 15 = 180. The same rule that expands 3(x + 5) is at work.

Suppose you buy x apples on Monday and 3x apples on Tuesday. Together you bought 4x apples: 1 + 3 = 4 groups of the same thing. Terms with exactly the same variable part are like terms. 5y and 2y are like terms. 5y and 2 are not, and 5y and 2y^2 are not, because the exponents differ. To simplify an expression, combine its like terms by adding or subtracting the coefficients.

Simplify 4a + 7 + 2a − 3. The like terms are 4a and 2a, which combine to 6a. The constants 7 and −3 combine to 4. The result is 6a + 4. Check with a = 1: the original is 4 + 7 + 2 − 3 = 10, and the simplified form is 6 + 4 = 10. Now simplify 3(x + 2) + 5x. First distribute: 3x + 6 + 5x. Then combine: 8x + 6.

The most common mistake is adding unlike terms, turning 6a + 4 into 10a. Try a = 2: 6a + 4 = 16, but 10a = 20. Another mistake is losing a sign. In 9n − 4n − 2n, subtract each time: 9 − 4 − 2 = 3, so 3n. Keep each sign attached to the term that follows it, and move terms carefully.

Expanding turns 3(x + 5) into 3x + 15. Factoring goes the other way: it turns 3x + 15 back into 3(x + 5). To factor, find the greatest common factor (GCF) of the terms, the largest number that divides all of them. For 3x + 15, the GCF of 3 and 15 is 3. Divide each term by 3: 3x ÷ 3 = x and 15 ÷ 3 = 5. Write the GCF outside and the leftovers inside: 3(x + 5).

Factor 12n + 18. The factors of 12 are 1, 2, 3, 4, 6, 12; the factors of 18 are 1, 2, 3, 6, 9, 18. The GCF is 6. Then 12n ÷ 6 = 2n and 18 ÷ 6 = 3, so 12n + 18 = 6(2n + 3). Check by expanding: 6 × 2n = 12n and 6 × 3 = 18. Factor 8y − 20: the GCF of 8 and 20 is 4, so 8y − 20 = 4(2y − 5).

Two errors are common. One is using a common factor that is not the greatest: 2(6n + 9) is true but not fully factored, because 6n + 9 still has a common factor of 3. The other is dropping a term: 12n + 18 is not 6(2n), since expanding gives 12n with no 18. Always expand your answer to check that every term comes back.

A rectangle has a length of x cm and a width of 4 cm. Its perimeter is the distance around it: x + 4 + x + 4. Combine like terms: 2x + 8 cm. You can also write 2(x + 4), which expands to the same 2x + 8. The rectangle’s area is length times width: 4x square cm. For x = 7, the perimeter is 2(7) + 8 = 22 cm and the area is 4 × 7 = 28 square cm.

Try a shape with more sides. A triangle has sides of n, n + 2 and 2n − 1. Its perimeter is n + (n + 2) + (2n − 1). Drop the parentheses (they follow plus signs, so nothing changes) and combine: n + n + 2n = 4n, and 2 − 1 = 1. Perimeter: 4n + 1. For n = 5, the sides are 5, 7 and 9, and 5 + 7 + 9 = 21. The expression gives 4(5) + 1 = 21. The check matches.

A square with side s has perimeter 4s and area s^2. Do not mix them up: for s = 6, the perimeter is 24 and the area is 36. Now picture a square garden of side s with a 2-foot path added along one full side. The region is s wide and s + 2 long, so its area is s(s + 2) = s^2 + 2s. For s = 10, that is 100 + 20 = 120 square feet, and 10 × 12 = 120 agrees.

A pattern of toothpick squares uses 4 toothpicks for 1 square, 7 for 2 squares in a row, 10 for 3, and 13 for 4. Each new square adds 3 toothpicks, since one side is shared with the square before it. To find the number for 50 squares, you could keep adding 3, or you could write an expression. Call the number of squares n. The count starts at 4 and adds 3 for each square after the first: 4 + 3(n − 1). Simplify: 4 + 3n − 3 = 3n + 1.

Test the nth term expression 3n + 1 against the pattern. For n = 1: 4. For n = 2: 7. For n = 3: 10. For n = 4: 13. It works, so for 50 squares you need 3(50) + 1 = 151 toothpicks. The coefficient 3 is the amount added each step. The constant 1 is what the pattern would be at step 0, before any square is built.

Here is a quicker way to build the expression for any sequence that grows by the same amount. Find the step size by subtracting neighbors: 7 − 4 = 3. Then find the constant by asking what number plus 3 × 1 gives the first term: 4 − 3 = 1. So the expression is 3n + 1. For the sequence 5, 9, 13, 17, the step is 4 and the constant is 5 − 4 = 1, so the nth term is 4n + 1. Check n = 3: 12 + 1 = 13. A common error is writing n + 4, which only says "add 4 each time" instead of giving any term directly.

An equation is a statement that two expressions are equal, written with an equals sign: 3 + n = 11. It asks a question: which number makes both sides the same? A solution is a number that makes the equation true. Try n = 7: 3 + 7 = 10, not 11, so 7 is not a solution. Try n = 8: 3 + 8 = 11. True, so the solution is 8. To solve an equation means to find every solution.

Picture a balance scale. The left pan holds a sealed bag of marbles and 3 loose marbles; the right pan holds 11 marbles; the scale is level. The bag is the variable. Whatever you do to one pan, you must do to the other, or the scale tips. Remove 3 marbles from each pan. Now the bag alone balances 8 marbles. That is the whole method of solving equations: keep the balance while making the variable stand alone.

Checking a solution means substituting it back and comparing both sides. For 2x − 5 = 9, a student claims x = 7. Check: 2(7) − 5 = 14 − 5 = 9. Both sides equal 9, so the answer is right. Another student claims x = 2. Check: 2(2) − 5 = −1, not 9. The check catches the error before it counts. The equals sign is not a signal to "write the answer." It is a promise that the two sides match.

In x + 9 = 15, the 9 is added to x. To get x alone, undo the addition by subtracting 9, and subtract it from both sides so the balance holds: x + 9 − 9 = 15 − 9, so x = 6. Check: 6 + 9 = 15. Subtraction and addition are inverse operations: each undoes the other. An equation solved with one inverse operation is a one-step equation.

In y − 12 = 20, the 12 is subtracted from y. Undo it by adding 12 to both sides: y = 32. Check: 32 − 12 = 20. Negative numbers change nothing about the method. In n + 7 = 3, subtract 7 from both sides: n = 3 − 7 = −4. Check: −4 + 7 = 3. In t − 5 = −2, add 5 to both sides: t = −2 + 5 = 3. Check: 3 − 5 = −2.

Two mistakes to avoid. One is doing the operation on only one side. From x + 9 = 15, writing x = 15 forgets that 9 was removed only from the left. The other is choosing the wrong inverse. Subtracting 12 from both sides of y − 12 = 20 gives y − 24 = 8. That is true, but no closer to the answer. Ask, "What is being done to the variable?" and do the opposite.

A situation: Deshawn had some money, spent $18 on a Chicago Bulls cap, and has $27 left. If m is what he started with, then m − 18 = 27. Add 18 to both sides: m = 45. Check: 45 − 18 = 27. He started with $45.

In 4x = 28, the 4 multiplies x. Undo it by dividing both sides by 4: x = 28 ÷ 4 = 7. Check: 4 × 7 = 28. In n/5 = 6, the n is divided by 5. Undo it by multiplying both sides by 5: n = 30. Check: 30 ÷ 5 = 6. Multiplication and division are inverse operations, just as addition and subtraction are.

The coefficient may be a fraction or a decimal. In 0.5y = 9, divide both sides by 0.5: y = 18. Check: 0.5 × 18 = 9. In (2/3)k = 10, multiply both sides by the reciprocal 3/2: k = 10 × 3/2 = 15. Check: (2/3) × 15 = 10. A negative coefficient works the same way. In −3m = 21, divide both sides by −3: m = −7. Check: −3 × (−7) = 21.

The most common mistake is subtracting the coefficient: from 4x = 28 writing x = 24. The 4 is multiplied, not added, so subtraction does not undo it. The check shows 4 × 24 = 96, far from 28. A second mistake is dividing only the side with the variable. Do the same operation to both sides every time, and finish with a check.

A situation: Ana split 42 stickers equally among her friends, and each friend got 6. If f is the number of friends, then 6f = 42, because 6 stickers times f friends makes 42. Divide both sides by 6: f = 7. Check: 6 × 7 = 42. Seven friends.

Maya’s phone bill is 20 + 5g. One month the bill is $55. How many gigabytes did she use? The equation is 20 + 5g = 55. Two things happen to g: it is multiplied by 5, then 20 is added. To undo them, work in reverse order. First subtract 20 from both sides: 5g = 35. Then divide both sides by 5: g = 7. Check: 20 + 5(7) = 20 + 35 = 55. She used 7 gigabytes.

This is a two-step equation. Think of putting on socks and then shoes; to undo, you take off the shoes first. Solve 3x − 4 = 17. Add 4 to both sides: 3x = 21. Divide by 3: x = 7. Check: 21 − 4 = 17. Solve n/2 + 6 = 10. Subtract 6: n/2 = 4. Multiply by 2: n = 8. Check: 4 + 6 = 10.

Negative numbers and fractions follow the same steps. Solve −2y + 5 = 11. Subtract 5: −2y = 6. Divide by −2: y = −3. Check: −2(−3) + 5 = 6 + 5 = 11. Solve (3/4)k − 1 = 5. Add 1: (3/4)k = 6. Multiply by 4/3: k = 8. Check: (3/4)(8) − 1 = 6 − 1 = 5.

The common mistake is dividing before subtracting. From 20 + 5g = 55, some students divide only the 55 by 5. They write 20 + g = 11, which is wrong. Dividing every term by 5 would give 4 + g = 11, which is correct, but it is easy to miss a term. The safe habit: undo the addition or subtraction first, then the multiplication or division.

Two plans: Maya’s costs 20 + 5g and a rival plan costs 32 + 2g. When do they cost the same? Set them equal: 20 + 5g = 32 + 2g. The variable appears on both sides. First collect the variable terms on one side by subtracting 2g from both sides: 20 + 3g = 32. Now it is a two-step equation. Subtract 20: 3g = 12. Divide by 3: g = 4. Check both sides: 20 + 20 = 40 and 32 + 8 = 40. At 4 gigabytes the plans match.

Choose the side that keeps the coefficient positive when you can; it avoids sign errors. Solve 7x − 3 = 4x + 12. Subtract 4x from both sides: 3x − 3 = 12. Add 3: 3x = 15. Divide: x = 5. Check: 35 − 3 = 32 and 20 + 12 = 32. Solve 2n + 9 = 5n − 6. Subtract 2n: 9 = 3n − 6. Add 6: 15 = 3n. Divide: n = 5. Check: 2(5) + 9 = 19 and 5(5) − 6 = 19. A careless check like 2 + 9 = 11 forgets to multiply 2 × 5 first, so always substitute before you add.

Common mistake: subtracting a variable term from one side only. From 7x − 3 = 4x + 12, a student writes 3x − 3 = 4x + 12. That keeps the 4x it was supposed to remove. Another mistake is combining a variable term with a constant. The expression 3x − 3 is not 0 or 3, because 3x and 3 are unlike terms. Work one operation per line, always on both sides, and finish with a check on the original equation.

Four friends each pay the same amount for a ticket and a $3 snack, and the total is $60. If t is the ticket price, then 4(t + 3) = 60. One route: distribute first, 4t + 12 = 60. Subtract 12: 4t = 48. Divide by 4: t = 12. Check: 4(12 + 3) = 4(15) = 60. Another route: divide both sides by 4 first, t + 3 = 15, then subtract 3, t = 12. Both routes work when the outside number divides the other side evenly.

Solve 3(2x − 5) + 4 = 25. Distribute: 6x − 15 + 4 = 25. Combine like terms: 6x − 11 = 25. Add 11: 6x = 36. Divide: x = 6. Check: 3(12 − 5) + 4 = 3(7) + 4 = 25. Solve 5(n + 2) = 2n + 22. Distribute: 5n + 10 = 2n + 22. Subtract 2n: 3n + 10 = 22. Subtract 10: 3n = 12. So n = 4. Check: 5(6) = 30 and 8 + 22 = 30.

The distribution mistake from the last chapter returns here. Writing 4(t + 3) as 4t + 3 leads to 4t = 57 and a messy wrong answer. A negative outside the parentheses needs extra care when you clear parentheses: −2(y − 3) is −2y + 6, not −2y − 6, because a negative times a negative is positive. Solve 10 − 2(y − 3) = 4. Distribute: 10 − 2y + 6 = 4. Combine: 16 − 2y = 4. Subtract 16: −2y = −12. Divide by −2: y = 6. Check: 10 − 2(3) = 4.

A sign on a ride says, "You must be at least 48 inches tall." That is not an equation. Many heights work. It is an inequality: h ≥ 48, read "h is greater than or equal to 48." There are four symbols. The symbol < means less than. The symbol > means greater than. The symbol ≤ means less than or equal to. The symbol ≥ means greater than or equal to. "At least" means ≥, "at most" means ≤, "more than" means >, and "fewer than" means <.

An inequality has a solution set, usually with infinitely many numbers in it. For h ≥ 48, the heights 48, 50, 61.5 and 72 all work; 47 does not. To graph it on a number line, put a closed circle at 48, because 48 itself is included. Then draw a ray to the right, toward larger numbers. For x < 3, put an open circle at 3, because 3 is not included. Then draw the ray to the left.

Read the direction carefully. Both 5 > x and x < 5 say the same thing: x is less than 5. Turn the statement around the symbol and the point of the arrow still aims at the smaller side. Common mistake: graphing x < 3 with the ray going right because "the arrow points right." Test a number: 0 is less than 3, and 0 is to the left of 3, so the ray goes left. Another slip is using a closed circle for < or >. Closed means "equal to" is allowed.

Solve x + 7 < 12 the way you solve an equation: subtract 7 from both sides, x < 5. Every number less than 5 is a solution. Solve 3n ≥ 21: divide both sides by 3, n ≥ 7. Solve 2y − 5 > 9: add 5, 2y > 14; divide by 2, y > 7. Adding, subtracting, and multiplying or dividing by a positive number all keep the inequality pointing the same way.

One rule is new. Start with a true statement: 2 < 6. Multiply both sides by −1 to get −2 and −6. But −2 is greater than −6, so the symbol must flip: −2 > −6. Whenever you multiply or divide both sides by a negative number, reverse the symbol. Solve −3m ≤ 12: divide both sides by −3 and flip, m ≥ −4. Use a check value such as m = 0: −3(0) = 0 ≤ 12 is true, and 0 ≥ −4 agrees.

Solve 8 − 2k > 2. Subtract 8: −2k > −6. Divide by −2 and flip: k < 3. Check with k = 1: 8 − 2 = 6 > 2, true. Check with k = 4: 8 − 8 = 0 > 2, false, and 4 < 3 is also false, so the answer holds. Common mistakes: flipping the symbol when you subtract a negative (you should not), and forgetting to flip when you divide by a negative (you must).

Maya wants her phone bill to stay at or under $45. With the bill 20 + 5g, that is 20 + 5g ≤ 45. Subtract 20: 5g ≤ 25. Divide by 5: g ≤ 5. In words: she can use at most 5 gigabytes. Gigabytes cannot be negative, so the useful answers are 0 through 5. Check the edge: 20 + 5(5) = 45, allowed. Check one beyond: 20 + 5(6) = 50, too much.

A school club has $200 and wants to buy T-shirts at $9 each after paying a $38 setup fee. How many shirts can they afford? The constraint is 38 + 9s ≤ 200. Subtract 38: 9s ≤ 162. Divide by 9: s ≤ 18. They can buy up to 18 shirts. Because shirts come in whole numbers, 18.5 would not be a choice even if the arithmetic allowed it. Check: 38 + 9(18) = 38 + 162 = 200, exactly the budget.

Sometimes the answer is a minimum. Jordan earns $12 an hour and needs more than $150 for a trip: 12h > 150. Divide by 12: h > 12.5. Since he is paid for whole hours, he must work at least 13 hours. The mistake to avoid is answering "12.5 hours" without asking whether that makes sense, or choosing 12 hours, which gives only $144. Always read the answer back against the story.

A story problem becomes an equation in four moves. Name the unknown with a letter. Translate the sentence into an equation. Solve. Answer in the story’s words and check. Story: "Three friends split a dinner bill equally after a $6 tip was added, and each pays $14." Let b be the bill before the tip. Then (b + 6)/3 = 14. Multiply by 3: b + 6 = 42. Subtract 6: b = 36. The bill was $36. Check: (36 + 6)/3 = 42/3 = 14.

Story: "A bus pass costs $5 per day, or a weekly pass costs $20" (example prices). After how many days does the daily total reach the weekly price? Let d be days: 5d = 20, so d = 4. On the fourth day the totals are equal, so a rider who rides 5 or more days saves with the weekly pass. Story: "The sum of three consecutive numbers is 48." Let the first be n; the next two are n + 1 and n + 2. Then n + (n + 1) + (n + 2) = 48, so 3n + 3 = 48, 3n = 45, and n = 15. The numbers are 15, 16 and 17.

Stories about lengths need the same care. "A rectangle’s length is 4 more than its width, and its perimeter is 32." Let w be the width; the length is w + 4. Perimeter: 2w + 2(w + 4) = 32, so 4w + 8 = 32, 4w = 24, and w = 6. Width 6, length 10. Check: 2(6) + 2(10) = 12 + 20 = 32. The common mistake is answering with w and forgetting that the question asked for the length, or for both.

A check is not extra credit; it is the last step of solving. Substitute the answer into the original equation, not into a later line where an error might already live. A student solves 5(x − 2) = 3x + 6 and gets x = 4. Check: 5(4 − 2) = 10 and 3(4) + 6 = 18. Not equal, so 4 is wrong. Redo: 5x − 10 = 3x + 6, so 2x = 16 and x = 8. Check: 5(6) = 30 and 24 + 6 = 30. Now it is right.

A check can also find a solution that fits the math but not the story. Suppose a problem about the number of people on a bus gives p = 7.5. Something is off, since people come in whole numbers. If a length comes out negative, the equation was set up with a sign backward. In an inequality problem, check an edge value and a value beyond the edge. For 20 + 5g ≤ 45, that means testing g = 5 and g = 6.

Some equations have surprising answers. Solve 2(x + 3) = 2x + 6. Distributing gives 2x + 6 = 2x + 6, which is true for every x, so every number is a solution; this is called an identity, and the two sides are equivalent expressions. Solve 2x + 1 = 2x + 5. Subtract 2x to get 1 = 5, which is never true, so the equation has no solution. When the variable disappears, look at what is left: a true statement means all numbers, a false one means none.