← SHSAT

Math: Expressions, Equations, and Inequalities

Writing an expression, rewriting it, solving it — and the word problems where most SHSAT math actually lives: consecutive integers, coins, mixture and work.

19 topics · 81 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

Writing Algebraic Expressions from Words

  1. How it worksHow do the words more, less, product and quotient map onto plus, minus, times and divide?
  2. How it worksHow does the phrase twice the sum of a number and 3 force parentheses into the expression?
  3. Worked exampleFind the expression for x shirts at 14 dollars each plus a flat 6 dollar shipping charge.
  4. Common mistakeWhy isn't the quotient of 12 and a number written as n divided by 12?
Your turneasy1 of 3
Write an expression for the total cost of x shirts at 16 dollars each plus a flat shipping charge of 9 dollars.

Evaluating Expressions

  1. How it worksHow does substituting a value inside parentheses protect the sign when the variable is negative?
  2. Worked exampleFind the value of 3x2 - 4x + 7 when x equals -2.
  3. Worked exampleFind the value of (a - b) divided by (a + b) when a is 7 and b is -3.
  4. Common mistakeWhy isn't 2x the same as x2 when plugging in x = 2 makes both of them 4?
Your turneasy1 of 3
Find the value of 3x2 - 4x + 7 when x equals negative 7.

Combining Like Terms

  1. How it worksHow does a variable behave like a unit so that 3 apples plus 5 apples stays in apples?
  2. Worked exampleFind the simplified form of 7x - 4 + 2x2 - 9x + 11.
  3. Common mistakeWhy isn't 3x + 5x equal to 8x2 when you just added the x parts together?
Your turneasy1 of 3
Simplify 7x - 4 + 7x2 - 9x + 11.

The Distributive Property

  1. How it worksHow does an area model show a(b + c) splitting into ab plus ac?
  2. How it worksHow does a minus sign parked in front of parentheses flip every sign inside them?
  3. Worked exampleFind the expanded form of -3(2x - 5) + 4(x + 1) and collect the like terms.
  4. Common mistakeWhy doesn't distributing to only the first term inside the parentheses ever work out?
  5. Going furtherWhat happens to the area model when one of the two pieces being distributed is negative?
Your turnmedium1 of 3
Expand and collect like terms: -3(2x - 5) + 12(x + 1).

Factoring Linear Expressions

  1. How it worksHow does finding the greatest common factor of the terms undo a distribution?
  2. Worked exampleFind the factored form of 18x - 24 and check it by distributing straight back.
  3. Worked exampleFind the factored form of -2x + 10 with the negative pulled out in front.
  4. Going furtherWhat happens to a factored expression when the common factor is a fraction such as 1/3?
Your turneasy1 of 3
The expression 7x2 + 11x is factored as x times a second factor. Write that second factor, in terms of x.

Recognizing Equivalent Forms of an Expression

  1. How it worksHow does 5(3 + x)/9 split into 5 times 3 over 9 plus 5x over 9 without changing value?
  2. How it worksHow does substituting one number into two expressions test whether they are equivalent?
  3. Worked exampleFind three different-looking expressions all equal to 5(3 + x)/9 and show why each matches.
  4. Worked exampleFind which of 2(x + 6), 2x + 12 and 2x + 6 refuses to match the other two.
  5. Common mistakeWhy isn't an expression written in a new shape automatically a different expression?
Your turneasy1 of 3
Find the value of 5(3 + x)/9 when x equals 17.

Solving One-Step Equations

  1. How it worksHow does the inverse operation peel away whatever is attached to the variable?
  2. Worked exampleFind x in x - 17 = -5 and then in -4x = 52, checking both answers.
  3. Common mistakeWhy doesn't a number keep the sign it had when it crosses the equals sign?
Your turneasy1 of 3
Solve for x: x - 17 = negative 11.

Solving Two-Step Equations

  1. How it worksHow does peeling operations off in reverse order unwrap the variable in exactly two moves?
  2. Worked exampleFind x in 5x - 8 = 37, watching the balance after every move.
  3. Worked exampleFind x in x/3 + 7 = 2 and check the answer back in the original equation.
  4. Common mistakeWhy doesn't dividing only the 5x by 5 and leaving the 8 alone keep the equation true?
Your turneasy1 of 3
Solve 7x - 8 = r for x, writing your answer in terms of r.

Solving Multi-Step Equations

  1. How it worksHow does clearing parentheses and collecting terms shrink a long equation into a two-step one?
  2. How it worksHow does multiplying every term by a common denominator wipe the fractions out of an equation?
  3. Worked exampleFind x in 3(2x - 4) + 5 = 2x + 9.
  4. Worked exampleFind x in (x + 2)/4 - (x - 1)/3 = 1 by clearing the denominators first.
  5. Common mistakeWhy doesn't handing the 3 to only the 2x in 3(2x - 4) leave an equivalent equation?
Your turnmedium1 of 3
Solve 3(2x - 4) + 5 = 4x + c for x, writing your answer in terms of c.

Setting Up an Equation from a Word Problem

  1. How it worksHow do you pick which quantity to call x when each one is described using the other?
  2. How it worksHow does every sentence of a word problem become one more piece of a single equation?
  3. Worked exampleFind how many marbles Jenny began with when she had twice Keiko's and led by 10 after giving away 5.
  4. Worked exampleFind the number when 4 less than three times it equals that number increased by 18.
  5. Ties togetherHow does a marble problem and a money problem end up wearing the same two-step equation?
Your turnmedium1 of 3
Four less than 7 times a number equals that number increased by c. Write an expression for the number, in terms of c.

Consecutive Integer Problems

  1. How it worksHow do three tiles labelled n, n + 1 and n + 2 stack into a total you can solve?
  2. How it worksHow does a run of consecutive even integers get written as n, n + 2 and n + 4?
  3. Worked exampleFind two consecutive integers summing to -15, then the product after adding 1 to the smaller and taking 2 from the larger.
  4. Worked exampleFind three consecutive odd integers that add up to 141.
  5. Common mistakeWhy doesn't n, n + 1, n + 2 describe consecutive even numbers when they still come one after another?
Your turnmedium1 of 3
Three integers, each 7 more than the one before it, have a sum of S. Write an expression for the smallest of the three, in terms of S.

Age Problems

  1. How it worksHow does a two-column table of ages now and ages later turn a puzzle into algebra?
  2. Worked exampleFind Maya's age when she is 4 times her brother and in 6 years will be only twice his age.
  3. Worked exampleFind both ages when a father is 3 times as old as his son and their ages total 48.
  4. Common mistakeWhy isn't the gap between two people's ages something that grows as the years pass?
Your turnmedium1 of 3
Maya is 4 times as old as her brother. In 2 times 9 years she will be only twice his age. How old is Maya now?

Coin and Value Problems

  1. How it worksHow does multiplying each coin count by its value turn a pile of coins into a money equation?
  2. Worked exampleFind how many dimes sit in a jar of 32 nickels and dimes worth 2.45 dollars.
  3. Worked exampleFind how many 8 dollar tickets sold when 150 tickets at 8 and 12 dollars brought in 1480.
  4. Common mistakeWhy isn't 5n + 10d = 2.45 the right equation for a jar holding 2.45 dollars?
Your turneasy1 of 3
A jar holds n nickels and 31 more dimes than nickels. Write an expression for the total value of the jar in dollars, in terms of n.

Mixture Problems

  1. How it worksHow does tracking the pure ingredient apart from the total turn a mixture into an equation?
  2. Worked exampleFind how much 40 percent acid to pour into 6 liters of 10 percent acid to reach 20 percent.
  3. Worked exampleFind the price per pound of a blend of 12 pounds of 6 dollar coffee and 8 pounds of 11 dollar coffee.
  4. Going furtherWhat happens to a mixture equation when one of the two ingredients is plain water?
Your turnhard1 of 3
How many liters of 40 percent acid must be poured into 6 liters of 10 percent acid to produce a mixture that is 25 percent acid?

Work Problems

  1. How it worksHow does thinking in jobs per hour instead of hours per job let two workers simply add?
  2. Worked exampleFind how long two pipes take together when one fills a tank in 6 hours and the other in 12.
  3. Worked exampleFind how long the second worker needs alone when two finish in 4 hours and one alone takes 12.
  4. Common mistakeWhy doesn't averaging two workers' solo times give the time they need side by side?
Your turneasy1 of 3
One pipe fills a tank in 10 hours and a second pipe fills it in h hours. Write an expression, in terms of h, for the number of hours the two take running together.

Writing and Solving Linear Inequalities

  1. How it worksHow do at least, at most and no more than turn into the four inequality symbols?
  2. How it worksHow does multiplying both sides by a negative spin the inequality symbol around?
  3. Worked exampleFind the sales needed when pay is 50 dollars plus 3 per sale and must reach at least 100.
  4. Worked exampleFind the solution of -2x + 7 > 15 and mark the exact moment the symbol flipped.
  5. Common mistakeWhy doesn't the symbol flip when you merely add a negative number to both sides?
Your turnmedium1 of 3
Solve -2x + 7 > 32 for x. The solution is every x below one boundary value. What is that boundary value?

Graphing an Inequality Solution Set on a Number Line

  1. How it worksHow does a shaded ray running right from 4 say that every number past 4 works?
  2. Worked exampleFind and graph the solution of 3x - 5 is less than or equal to 7 on a number line.
  3. Worked exampleFind the inequality drawn as a filled dot at -2 with shading stretching left forever.
  4. Common mistakeWhy doesn't a solution ray stop where the drawn number line runs out of room?
Your turneasy1 of 3
Solve 3x - 5 is less than or equal to 30. What is the boundary value where the graph's dot is drawn?

Systems of Equations Solved by Substitution

  1. How it worksHow does solving one equation for a variable let that expression slide into the other one?
  2. How it worksHow does the answer to a system appear as the single point where two lines cross?
  3. Worked exampleFind x and y when y = 2x - 3 and 4x + y = 15.
  4. Common mistakeWhy doesn't a value that satisfies one equation of a system automatically solve the system?
  5. Worked exampleFind two numbers whose sum is 54 when one of them is 6 less than twice the other.
Your turneasy1 of 3
Solve the system y = 2x - 3 and 7x + y = c. Write an expression for x, in terms of c.

Ratio Problems That Cross Into Geometry

  1. How it worksHow does naming the sides 3x and 2x turn a perimeter fact into a one-variable equation?
  2. Worked exampleFind the dimensions of a rectangle with perimeter 510 centimeters and length to width 3 to 2.
  3. Worked exampleFind the three angles of a triangle whose measures sit in the ratio 2 to 3 to 4.
  4. Going furtherWhat happens to the area of a rectangle when both sides of a 3 to 2 ratio are doubled?
Your turnmedium1 of 3
A rectangle has perimeter P centimeters and a length to width ratio of 7 to 2. Write an expression for its length, in terms of P.