← SSAT® Upper Level

Algebra: Expressions, Equations and Exponents

Variables in equations and inequalities, the exponent rules, absolute value and the algebraic word problems E3n lists under Algebra, Geometry and Other Math.

9 topics · 37 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.

Variables, Expressions and Substitution

  1. How it worksHow does substituting a value for x collapse a whole expression down to one number?
  2. Worked exampleFind the value of 3a² - 2ab when a = -2 and b = 5.
  3. Worked exampleWalk me through a made-up operation where a # b means 2a + 3b, and find 4 # 5.
  4. Common mistakeWhy isn't 3x the same thing as x³ when both are built from a 3 and an x?
Your turneasy1 of 3
Find the value of 3a2 - 2ab when a = -4 and b = 7.

Like Terms and the Distributive Property

  1. How it worksHow does distributing a 5 across (x + 7) hand that 5 to both terms and not just the first?
  2. Worked exampleFind the simplified form of 4(2x - 3) - 2(x - 5).
  3. Common mistakeWhy doesn't -(x - 4) simplify down to -x - 4?
Your turnmedium1 of 3
Simplify 4(2x - 3) - 2(x - 7) into a single expression in x.

Solving Linear Equations in One Variable

  1. How it worksHow does undoing an equation in the reverse of the order of operations strand x by itself?
  2. Worked exampleWalk me through solving 5(x - 3) = 2x + 9 the way the SSAT wants it done in under a minute.
  3. Common mistakeWhy doesn't a term keep its original sign when it crosses the equals sign?
  4. Worked exampleFind x when (x + 4)/3 = 7 - x.
  5. Going furtherWhat happens when solving an equation ends in 5 = 5, and what happens when it ends in 5 = 7?
Your turneasy1 of 3
Solve 7(x - 3) = 2x + c for x, writing your answer in terms of c.

Inequalities and When the Sign Flips

  1. How it worksHow does multiplying both sides by a negative number spin the inequality symbol around?
  2. Worked exampleFind every value of x satisfying -3x + 5 > 14 and shade them on a number line.
  3. Common mistakeWhy isn't x > 4 the same shaded stretch of the number line as 4 > x?
  4. Going furtherWhat changes about the solution set when x is also required to be a whole number?
Your turnmedium1 of 3
The solutions of -7x + 5 > c are all the values of x below one boundary number. Write an expression for that boundary, in terms of c.

Properties of Exponents

  1. How it worksHow do the repeated factors on the top and bottom of x⁵/x² literally cancel away to leave x³?
  2. Common mistakeWhy isn't x² + x³ equal to x⁵ when x² times x³ certainly is?
  3. Worked exampleFind the value of (2³ × 2⁵) ÷ 2⁶ without working out any of those powers separately.
  4. Worked exampleWalk me through simplifying (3x²y)³ so that every factor inside the parentheses gets cubed.
  5. Going furtherWhat do a zero exponent and a negative exponent mean once the cancelling picture is in front of you?
Your turneasy1 of 3
Simplify (x5 times x6) divided by x5, writing the result as a single power of x.

Square Roots and Perfect Squares

  1. How it worksHow does pulling perfect squares out of the root of 72 turn it into 6 times the root of 2?
  2. Worked exampleFind the root of 144 plus the root of 25 minus the root of 81 using squares you know by heart.
  3. Worked exampleWalk me through estimating the square root of 50 to the nearest tenth with no calculator.
Your turneasy1 of 3
Using squares you know by heart, what is √(172) + √25 - √81?

Absolute Value

  1. How it worksHow does reading the bars around x as distance from zero explain why the result is never negative?
  2. Worked exampleFind both solutions of the equation with 8 as the absolute value of x - 3, by walking 8 units each way.
  3. Worked exampleFind the value of the absolute value of 3 - 11, plus that of -5, minus that of 2 - 2.
  4. Going furtherWhat does a solution set of numbers less than 5 units from zero look like as a stretch of number line?
  5. Common mistakeWhy isn't the absolute value of a sum always equal to the sum of the two absolute values?
Your turneasy1 of 3
What is abs(3 - 24) + abs(-5) - abs(2 - 2)?

Algebraic Word Problems

  1. How it worksHow does five less than three times a number become 3n - 5 and never 5 - 3n?
  2. Worked exampleWalk me through a problem where Maya is 4 years older than Sam and their ages add to 38.
  3. Worked exampleFind the two consecutive even integers that add up to 74.
  4. Worked exampleFind how many dimes there are when 20 dimes and quarters together are worth 3.65 in dollars.
  5. Common mistakeWhy doesn't the quotient of 8 and x mean x divided by 8?
Your turneasy1 of 3
Maya is 7 years older than Sam and their ages add to t. Write an expression for Sam's age, in terms of t.

Two Equations, Two Unknowns

  1. How it worksHow does adding two equations together make one of the variables vanish outright?
  2. Worked exampleFind x and y when x + y = 14 and x - y = 4.
  3. Worked exampleWalk me through solving 2a + 3b = 19 together with a = b + 2 by substitution.
Your turneasy1 of 3
Solve the pair x + y = s and x - y = 9. Write an expression for x, in terms of s.

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