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Quantitative Reasoning: Algebra

Chapter 2 of the ETS Math Review end to end - expressions, exponents, linear and quadratic solving, inequalities, functions, word problems and the xy-plane.

29 topics · 116 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.

Algebraic Expressions: Terms, Coefficients and Degree

  1. How it worksHow do you read the coefficient and the degree off a term like negative 7x times y cubed?
  2. Worked exampleFind the number of terms and the degree of 4x to the 6th plus 7x to the 5th minus 3x plus 2.
  3. Common mistakeWhy isn't the degree of a constant term such as 5 equal to 1?
Your turneasy1 of 3
Find the degree of the term -7x5y5.

Combining Like Terms and Multiplying Expressions

  1. How it worksHow does multiplying every term of one expression by every term of the other cover all the products?
  2. How it worksWalk me through multiplying x plus 2 by 3x minus 7, term by term.
  3. Worked exampleFind the simplified form of 2wz plus 5z squared minus z squared plus 6wz.
  4. Common mistakeWhy isn't the square of x plus 3 equal to x squared plus 9?
Your turneasy1 of 3
Write 7wz + 9z2 - z2 + 9wz in simplified form.

Factoring Out a Common Factor

  1. How it worksHow do you spot the largest factor shared by every term of 15y squared minus 9y?
  2. Common mistakeWhy isn't 4x plus 12 equal to 4 times the quantity x plus 12?
  3. Worked exampleFind the factored form of 15y squared minus 9y and check it by multiplying back out.
Your turneasy1 of 3
The expression 9y2 + 10y is written as y times another expression. Write that other expression in terms of y.

Difference of Squares and Perfect Square Identities

  1. How it worksHow does a big square with a smaller square cut out of the corner prove the difference of squares?
  2. How it worksWalk me through why a plus b, all squared, picks up a middle term of 2ab.
  3. Worked exampleFind the value of 101 times 99 in your head using the difference of squares.
  4. Common mistakeWhy isn't a minus b, all squared, equal to a squared minus b squared?
  5. Going furtherWhat happens to the difference of squares pattern when the two squares are x to the 4th and 9?
Your turneasy1 of 3
Find the value of (145 + 1)(145 - 1) without multiplying the two numbers out directly.

Simplifying Rational Expressions

  1. How it worksHow does factoring the numerator and the denominator first make the cancelling legal?
  2. Common mistakeWhy isn't the quantity x plus 2, divided by 2, equal to x?
  3. Worked exampleFind the simplified form of x squared plus 9x plus 14 over 2x plus 4, and say which x value it excludes.
  4. Going furtherWhat happens to the domain of a rational expression after you cancel the factor that could be zero?
Your turnmedium1 of 3
Write (x2 - 92) divided by (x + 9) in simplified form.

Rules of Exponents

  1. How it worksHow do the repeated factors literally cancel when you divide x to the 7th by x to the 4th?
  2. Worked exampleFind the value of 2 to the 3rd times 2 to the 4th, divided by 2 to the 5th.
  3. Worked exampleFind the simplified form of the quantity 3x squared y cubed, raised to the 3rd power.
  4. Common mistakeWhy isn't x squared plus x cubed equal to x to the 5th?
  5. Going furtherWhat happens with a power of a power, such as x squared raised to the 4th, and why do exponents multiply?
Your turneasy1 of 3
Find the value of 26 times 25 divided by 26.

Solving Linear Equations in One Variable

  1. How it worksHow do you unwind an equation by undoing operations in the reverse order they were applied?
  2. Worked exampleWalk me through solving 3 times the quantity x minus 2 equals 5x plus 8 in under 40 seconds.
  3. Common mistakeWhy isn't dividing both sides by x a safe move when you are solving x squared equals 3x?
  4. Worked exampleFind x when 11x minus 4 minus 8x equals 2x plus 3.
Your turneasy1 of 3
Find x when 27x - 7 - 8x = 3x + 2.

Simultaneous Linear Equations

  1. How it worksHow does substitution collapse two equations in two unknowns down to one equation in one unknown?
  2. How it worksWalk me through elimination, adding two equations so that one of the variables disappears.
  3. Worked exampleFind x and y when 4x plus 3y equals 13 and x plus 2y equals 2.
  4. Common mistakeWhy isn't solving one of the two equations for y enough to pin down a single pair of values?
  5. Going furtherWhat happens when a system hands you three equations but only two unknowns?
Your turnmedium1 of 3
Find x when 5x + 4y = 39 and x + 7y = 2.

Systems with No Solution or Infinitely Many

  1. How it worksHow does elimination announce an inconsistent system by producing a false statement like 0 equals 5?
  2. Going furtherWhat happens when two equations turn out to be the same line written two different ways?
  3. Common mistakeWhy isn't a system of two equations in two unknowns guaranteed to have exactly one solution?
Your turnhard1 of 3
The system 7x + 9y = 19 and 7x + ky = 38 fails to have exactly one solution for one value of k. Find that value of k.

Solving Quadratic Equations by Factoring

  1. How it worksHow does a product being zero only when a factor is zero turn factoring into solving?
  2. Worked exampleFind both solutions of x squared minus 2x minus 3 equals 0 by factoring.
  3. Common mistakeWhy isn't 3 the only solution of x squared equals 3x?
  4. Going furtherWhat happens when a quadratic factors into two identical binomials?
Your turnmedium1 of 3
Find the larger solution of x2 - (5 + 13)x + 5 times 13 = 0.

The Quadratic Formula and the Discriminant

  1. How it worksHow does the plus or minus inside the quadratic formula produce both roots at once?
  2. Common mistakeWhy isn't a quadratic with no real solutions the same as a quadratic whose solution is zero?
  3. Worked exampleFind the solutions of x squared plus 4x minus 21 equals 0 using the quadratic formula.
  4. Going furtherWhat happens to the roots when the expression under the square root sign turns negative?
  5. Ties togetherHow does the sign of the discriminant match the number of times a parabola crosses the x-axis?
Your turnmedium1 of 3
Find the larger solution of x2 + 8x - 28 = 0.

Completing the Square

  1. How it worksHow does adding and then subtracting the same number turn a quadratic into a square plus a constant?
  2. Worked exampleFind the vertex form of x squared plus 6x plus 5 by completing the square.
  3. Going furtherWhat happens to completing the square when the number in front of x squared is not 1?
Your turnmedium1 of 3
Find the minimum value of y = x2 + 15x + 10.

Solving Linear Inequalities

  1. How it worksHow does multiplying both sides by a negative number flip the inequality sign right around?
  2. Worked exampleFind every value of x satisfying 4x plus 1 less than or equal to 7.
  3. Common mistakeWhy isn't dividing both sides of an inequality by x allowed when the sign of x is unknown?
  4. Common mistakeWhy doesn't adding a negative number to both sides of an inequality flip the sign?
  5. Going furtherWhat happens to a solution set on the number line when both sides get multiplied by negative 3?
Your turneasy1 of 3
Find the largest value of x satisfying 7x + 7 less than or equal to 57.

Compound and Absolute Value Inequalities

  1. How it worksHow does an absolute value inequality split into two ordinary inequalities joined by and, or by or?
  2. Going furtherWhat happens to the solution set when absolute value of x is greater than 5 rather than less than 5?
  3. Worked exampleFind every x satisfying absolute value of 2x minus 6 less than 4, and draw the result on a number line.
  4. Common mistakeWhy isn't absolute value of x greater than negative 2 an inequality with a real cutoff?
Your turnmedium1 of 3
Find the largest value of x satisfying abs(2x - 15) less than or equal to 7.

Simultaneous Inequalities

  1. How it worksHow do you add two inequalities together safely, and why can you never subtract one from the other?
  2. Common mistakeWhy isn't the largest value of x minus y found by subtracting the two given ranges directly?
  3. Worked exampleFind the possible range of x plus y when x lies between 2 and 5 and y lies between negative 1 and 3.
Your turneasy1 of 3
x lies strictly between 3 and 8, and y lies strictly between -1 and 3. Find the value that x + y stays below.

Functions: Inputs, Outputs and Notation

  1. How it worksHow does f of x notation tell you what to substitute and exactly where to substitute it?
  2. Worked exampleFind f of negative 2 and f of 3 when f of x equals x squared minus 2x plus 3.
  3. Common mistakeWhy isn't f of a plus b equal to f of a plus f of b for most functions?
Your turneasy1 of 3
f(x) = x2 - 7x + 8. Find f(-2).

Domain and Range

  1. How it worksHow do a zero denominator and a negative under a square root carve holes out of a domain?
  2. Common mistakeWhy isn't the domain of a function always just all of the real numbers?
  3. Worked exampleFind the domain of 2x over x minus 6, and the domain of the square root of x plus 2.
  4. Going furtherWhat happens to a function's range once its domain is squeezed into a short interval?
Your turneasy1 of 3
Find the value of x excluded from the domain of 7x divided by (x - 11).

Translating Words into Equations

  1. How it worksHow do phrases like more than, of, per and the result translate into symbols one piece at a time?
  2. Worked exampleFind the expression for a taxi trip that costs 3 dollars plus 1 dollar 25 for every mile driven.
  3. Going furtherWhat happens to your equation when the quantity you named is not the quantity the question wants?
  4. Ties togetherHow does writing an odd integer as 2n plus 1 tie word problems back to number properties?
Your turneasy1 of 3
A taxi trip costs 7 dollars plus 5 dollars for every mile driven. Write the cost in terms of the number of miles m.

Average and Mixture Word Problems

  1. How it worksHow does turning an average back into a total sum make the algebra a single linear equation?
  2. Worked exampleFind the score needed on a fourth exam to average 85 after scores of 82, 74 and 90.
  3. Worked exampleFind how much oil to add to 12 grams of a 40 percent vinegar mixture to make it 25 percent vinegar.
  4. Common mistakeWhy isn't the average of two mixtures the same as the average of their two percentages?
  5. Going furtherWhat happens to a weighted average when one of the two groups is three times the size of the other?
Your turneasy1 of 3
A student has scores of 84, 83 and 97. Find the score needed on a fourth exam to bring the average to 83.

Simple and Compound Interest

  1. How it worksHow does compounding turn interest into repeated multiplication rather than repeated addition?
  2. Worked exampleFind the balance after 2 years on 10,000 invested at a simple annual rate of 6 percent.
  3. Common mistakeWhy isn't 3.5 percent compounded quarterly the same as 3.5 percent paid once a year?
  4. Going furtherWhat happens to the time an investment takes to double when the interest rate doubles?
Your turneasy1 of 3
Find the balance after 2 years on 15446 dollars invested at a simple annual rate of 7 percent.

Cost, Revenue and Profit Problems

  1. How it worksHow do a fixed cost and a per-unit cost combine into one linear equation for total cost?
  2. Worked exampleFind the break-even count when radios cost 30 each to build, fixed costs run 1,200, and each sells for 50.
  3. Common mistakeWhy isn't revenue the same thing as profit in a GRE word problem?
  4. Going furtherWhat happens to the break-even point when fixed costs rise but the selling price stays put?
Your turnmedium1 of 3
Radios cost 34 dollars each to build, fixed costs run 2298 dollars, and each radio sells for 78 dollars. Find the number of radios that must be sold to break even.

The xy-Plane, Quadrants and Distance

  1. How it worksHow do the signs of x and y sort a point into one of the four quadrants?
  2. How it worksWalk me through why the distance between two points is the Pythagorean theorem in disguise.
  3. Worked exampleFind the distance between the points (2, 3) and (-3, -5) in the xy-plane.
  4. Common mistakeWhy isn't a point's reflection across the x-axis the same as its reflection across the y-axis?
Your turneasy1 of 3
Find the distance between the points (7, 7) and (-8, -4) in the xy-plane.

Slope of a Line

  1. How it worksHow does rise over run between any two points on a line always give back the same value?
  2. Worked exampleFind the slope of the line through the points (-2, -4) and (3, 3).
  3. Common mistakeWhy isn't the slope of a vertical line equal to zero?
  4. Worked exampleFind the slope of 2y plus x equals 6 by rewriting it in slope-intercept form.
Your turneasy1 of 3
Find the slope of the line through (-4, -7) and (8, 3).

Slope-Intercept Form and Intercepts

  1. How it worksHow does setting y to zero find an x-intercept while setting x to zero finds a y-intercept?
  2. How it worksWalk me through building the equation of a line from one point on it and a known slope.
  3. Worked exampleFind the x-intercept and the y-intercept of the line 3x plus 4y equals 24.
  4. Common mistakeWhy isn't b in y equals mx plus b the place where the line crosses the x-axis?
  5. Going furtherWhat happens to a line's equation when the line is horizontal, and when it is vertical?
Your turneasy1 of 3
Find the x-intercept of the line 7x + 9y = 57.

Parallel and Perpendicular Lines

  1. How it worksHow does rotating a slope triangle a quarter turn swap the rise with the run and flip one sign?
  2. Common mistakeWhy isn't a line of slope 2 perpendicular to a line of slope negative 2?
  3. Worked exampleFind the equation of the line through (4, 1) that is perpendicular to y equals 2x plus 5.
Your turnhard1 of 3
Write, in terms of x, the equation of the line through (7, 6) that is perpendicular to y = 9x + 3, solved for y.

Graphing Linear Inequalities in the Plane

  1. How it worksHow does testing the origin tell you which side of the boundary line to shade in?
  2. Worked exampleGraph the region where y is less than 2x plus 1 and say whether the boundary is dashed or solid.
  3. Going furtherWhat happens to the shaded region when two linear inequalities have to hold at the same time?
Your turneasy1 of 3
Find the y-coordinate of the boundary line y = 7x + 10 at x = 11.

Parabolas: Vertex, Axis and Intercepts

  1. How it worksHow does the sign of the leading coefficient decide whether the vertex is the low point or the high point?
  2. Common mistakeWhy isn't the vertex of a parabola always sitting on the y-axis?
  3. Worked exampleFind the x-intercepts, the y-intercept and the vertex of y equals x squared minus 2x minus 3.
  4. Going furtherWhat happens to a parabola's two x-intercepts as the graph slides up until it just grazes the axis?
  5. Ties togetherHow does the axis of symmetry land exactly halfway between the two roots of the quadratic?
Your turnmedium1 of 3
Find the x-coordinate of the vertex of y = x2 - 15x - 10.

Circles in the xy-Plane

  1. How it worksHow do you read a circle's center and radius straight out of its squared-brackets equation?
  2. Worked exampleFind the center and the radius of the circle x minus 6, squared, plus y plus 5, squared, equals 9.
  3. Common mistakeWhy isn't x squared plus y squared equals 100 a circle whose radius is 100?
Your turneasy1 of 3
Find the radius of the circle (x - 7)2 + (y + 7)2 = 94.

Translating and Reflecting Graphs

  1. How it worksHow does replacing x with negative x flip an entire graph across the y-axis?
  2. Worked exampleGraph y equals negative one half times the quantity x plus 2, squared, starting from y equals x squared.
  3. Common mistakeWhy isn't the graph of f of x minus 2 the same as the graph of f of x pushed down by 2?
  4. Common mistakeWhy doesn't multiplying a whole function by 2 move its graph up by 2?
  5. Going furtherWhat happens to a graph when you reflect it across the line y equals x?
Your turnmedium1 of 3
Write, in terms of x, the function whose graph is y = x2 shifted 7 units to the left and 8 units down.

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