Road to 800

Rewriting expressions and equations

Chapter 11 · 5 lessons · 192 questions

Rational equations

Key points

  • A rational equation has the variable in a denominator.
  • If the denominators are the same, set the numerators equal.
  • If there is one fraction on each side, cross-multiply.
  • Reject any answer that makes a denominator 00.
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See it solved

Example

x+4x−2=3\dfrac{x + 4}{x - 2} = 3

What is the solution to the given equation?

Worked by hand

The plan

  1. Determine the approach
  2. Solve the equation
  3. Check for an extraneous solution

Desmos is quicker here: type the equation and read the vertical line. A rejected value never appears.

  1. Step 1: Determine the approach

    • One fraction equals a number. Multiply both sides by the denominator, x−2x - 2.
    • x+4=3(x−2)x + 4 = 3(x - 2).

  2. Step 2: Solve the equation

    • 3⋅x=3x3 \cdot x = 3x and 3⋅(−2)=−63 \cdot (-2) = -6. So x+4=3x−6x + 4 = 3x - 6.
    • Subtract xx from both sides: 4=2x−64 = 2x - 6.
    • Add 66 to both sides: 10=2x10 = 2x.
    • x=102=5x = \dfrac{10}{2} = 5.

  3. Step 3: Check for an extraneous solution

    • The denominator is x−2x - 2. At x=5x = 5 it is 5−2=35 - 2 = 3, which is not 00. So 55 is allowed.
    • Check: 5+45−2=93=3\dfrac{5 + 4}{5 - 2} = \dfrac{9}{3} = 3. It works.
The correct answer is 55.

The same example in Desmos

  1. new graph

    Keys: (x+4)/x-2 → =3

    One vertical line.

  2. read where the line meets the x-axis

    x=5x = 5.

Type
Desmos expression list: frac{x+4}{x-2}=3
Look
Desmos graph of frac{x+4}{x-2}=3, with (5, 0) labelled

Made with Desmos

Open Desmos

Your turn

x2x−3=9x−3\dfrac{x^2}{x - 3} = \dfrac{9}{x - 3}

What is the solution to the given equation?

Show the answerHide the answer

Answer

The correct answer is −3-3.

  1. Step 1: Determine the approach

    • The denominators are the same, so the numerators are equal: x2=9x^2 = 9.

  2. Step 2: Solve the equation

    • x=3x = 3 or x=−3x = -3.

  3. Step 3: Check for an extraneous solution

    • At x=3x = 3 the denominator x−3x - 3 is 00. Dividing by 00 is undefined, so 33 is rejected.
    • At x=−3x = -3 the denominator is −3−3=−6-3 - 3 = -6, which is fine.
    • The only solution is −3-3.

Traps

  • Multiplying only one term. 3(x−2)3(x - 2) is 3x−63x - 6, not 3x−23x - 2. With x+4=3x−2x + 4 = 3x - 2 you get x=3x = 3, a wrong answer.
  • Keeping a value that makes a denominator 00. In Your turn, x=3x = 3 solves x2=9x^2 = 9, but it makes x−3=0x - 3 = 0. It is not a solution.
Practice this lesson28 questions

Rewriting rational expressions

Key points

  • To simplify a fraction, factor the top and the bottom, then cancel a factor they share.
  • You can cancel factors, never terms. In x+3x\dfrac{x + 3}{x} nothing cancels.
  • To add or subtract fractions, write them over a common denominator.
  • When you subtract a fraction, the minus sign applies to its whole numerator.
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See it solved

Example

Which expression is equivalent to x2−4x2+5x+6\dfrac{x^2 - 4}{x^2 + 5x + 6}, where x≠−2x \ne -2 and x≠−3x \ne -3?

  1. Ax−2x+3\dfrac{x - 2}{x + 3}
  2. Bx+2x+3\dfrac{x + 2}{x + 3}
  3. C−45x+6\dfrac{-4}{5x + 6}
  4. Dx−2x+2\dfrac{x - 2}{x + 2}

Worked by hand

The plan

  1. Factor the numerator and the denominator
  2. Cancel the common factor

With answer choices, Desmos is quicker: graph the given expression, then each choice, and keep the one that lies exactly on top.

  1. Step 1: Factor the numerator and the denominator

    • The top, x2−4x^2 - 4, is a difference of squares: (x−2)(x+2)(x - 2)(x + 2).
    • For the bottom, two numbers that multiply to 66 and add to 55 are 22 and 33.
    • So x2+5x+6x^2 + 5x + 6 =(x+2)(x+3)= (x + 2)(x + 3).

  2. Step 2: Cancel the common factor

    • The fraction is (x−2)(x+2)(x+2)(x+3)\dfrac{(x - 2)(x + 2)}{(x + 2)(x + 3)}.
    • The top and the bottom both have the factor (x+2)(x + 2). Cancel it.
    • What is left is x−2x+3\dfrac{x - 2}{x + 3}.
The correct answer choice is A: x−2x+3\dfrac{x - 2}{x + 3}.

The same example in Desmos

  1. new graph

    Keys: y=(x^2 → -4)/x^2 → +5x+6

    The given expression, as a curve.

  2. Keys: y=(x-2)/x+3

    Choice A lies exactly on top of it (dashed in the picture).

Type
Desmos expression list: y=frac{x^2-4}{x^2+5x+6}; y=frac{x-2}{x+3}
Look
Desmos graph of y=frac{x^2-4}{x^2+5x+6}; y=frac{x-2}{x+3}

Made with Desmos

Open Desmos

Your turn

The expression 3x+2x+1\dfrac{3}{x} + \dfrac{2}{x + 1} is equivalent to ax+3x(x+1)\dfrac{ax + 3}{x(x + 1)}, where aa is a constant and x>0x > 0. What is the value of aa?

Show the answerHide the answer

Answer

The correct answer is 55.

  1. Step 1: Write both fractions over a common denominator

    • The common denominator is x(x+1)x(x + 1).
    • 3x=3(x+1)x(x+1)\dfrac{3}{x} = \dfrac{3(x + 1)}{x(x + 1)} and 2x+1=2xx(x+1)\dfrac{2}{x + 1} = \dfrac{2x}{x(x + 1)}.

  2. Step 2: Add the numerators

    • 3(x+1)+2x3(x + 1) + 2x =3x+3+2x= 3x + 3 + 2x.
    • 3x+2x=5x3x + 2x = 5x, so the top is 5x+35x + 3.
    • The sum is 5x+3x(x+1)\dfrac{5x + 3}{x(x + 1)}. So a=5a = 5.

Traps

  • Cancelling terms instead of factors. Crossing out the x2x^2 on the top and the bottom leaves −45x+6\dfrac{-4}{5x + 6}: choice C, a wrong answer. Factor first.
  • Cancelling the wrong factor. (x+2)(x + 2) is the factor on both the top and the bottom. Choice D keeps it and drops (x+3)(x + 3), a wrong answer.
Practice this lesson36 questions

Rearranging a formula

Key points

  • Find the target letter first. The last sentence usually names it.
  • Treat every other letter like a number.
  • Undo things in reverse order: clear a fraction, then divide, then add or subtract what is left.
  • If the target letter is squared, take the square root last.
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See it solved

Example

The formula A=12h(b+c)A = \dfrac{1}{2}h(b + c) gives the area of a trapezoid. Which equation gives bb in terms of AA, hh and cc?

  1. Ab=2Ah−cb = \dfrac{2A}{h} - c
  2. Bb=A2h−cb = \dfrac{A}{2h} - c
  3. Cb=2A−hcb = 2A - hc
  4. Db=2A−chb = \dfrac{2A - c}{h}

Worked by hand

The plan

  1. Clear the fraction
  2. Isolate the target letter

The algebra is the method. Desmos is a check: give each letter an easy number and see which choice returns the right value.

  1. Step 1: Clear the fraction

    • The target letter is bb.
    • Multiply both sides by 22: 2A=h(b+c)2A = h(b + c).

  2. Step 2: Isolate the target letter

    • Divide both sides by hh: 2Ah=b+c\dfrac{2A}{h} = b + c.
    • Subtract cc from both sides: 2Ah−c=b\dfrac{2A}{h} - c = b.
    • That is choice A.
The correct answer choice is A: b=2Ah−cb = \dfrac{2A}{h} - c.

The same example in Desmos

  1. new graph

    Three rows with easy numbers. Desmos puts a slider under each.

  2. Keys: A=1/2 → h(b+c)

    A=16A = 16.

  3. Keys: 2A/h → -c

    33, the value of bb. Choice A is right.

  4. Keys: A/2h → -c

    −3-3, not 33. Choice B is wrong.

Type
Desmos expression list: h=4; b=3; c=5; A=frac{1}{2}h(b+c); frac{2A}{h}-c; frac{A}{2h}-c

Made with Desmos

Open Desmos

Your turn

The formula E=12mv2E = \dfrac{1}{2}mv^2 relates the positive quantities EE, mm and vv. What is vv in terms of EE and mm?

Show the answerHide the answer

Answer

The correct answer is v=2Emv = \sqrt{\dfrac{2E}{m}}.

  1. Step 1: Clear the fraction

    • The target letter is vv.
    • Multiply both sides by 22: 2E=mv22E = mv^2.

  2. Step 2: Isolate the target letter

    • Divide both sides by mm: 2Em=v2\dfrac{2E}{m} = v^2.
    • Take the square root of both sides: v=2Emv = \sqrt{\dfrac{2E}{m}}.
    • vv is positive, so only the positive root is kept.

Traps

  • Subtracting before dividing. In 2A=h(b+c)2A = h(b + c), the cc is inside the parentheses, so you cannot subtract it first. 2A−ch\dfrac{2A - c}{h} is choice D, a wrong answer.
  • Dividing by 22 instead of multiplying. To undo 12\dfrac{1}{2}, multiply by 22. A2h−c\dfrac{A}{2h} - c is choice B, a wrong answer.
Practice this lesson28 questions

Expanding, combining and factoring

Key points

  • To multiply two binomials, multiply each term of the first by each term of the second (FOIL).
  • A minus sign in front of parentheses changes the sign of every term inside.
  • Like terms have the same letters with the same exponents. Only like terms can be combined.
  • Special products: (a+b)2(a + b)^2 =a2+2ab+b2= a^2 + 2ab + b^2, (a−b)2(a - b)^2 =a2−2ab+b2= a^2 - 2ab + b^2 and (a+b)(a−b)(a + b)(a - b) =a2−b2= a^2 - b^2.
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See it solved

Example

Which expression is equivalent to (2x+3)(x−4)−(x2−5)(2x + 3)(x - 4) - (x^2 - 5)⁠?

  1. Ax2−5x−7x^2 - 5x - 7
  2. Bx2−5x−17x^2 - 5x - 17
  3. Cx2+5x−7x^2 + 5x - 7
  4. D3x2−5x−173x^2 - 5x - 17

Worked by hand

The plan

  1. Simplify the parentheses
  2. Combine like terms

The algebra is the method. Desmos is a check: graph the given expression and a choice. Equivalent expressions lie exactly on top of each other.

  1. Step 1: Simplify the parentheses

    • FOIL (2x+3)(x−4)(2x + 3)(x - 4). First 2x⋅x=2x22x \cdot x = 2x^2 and 2x⋅(−4)=−8x2x \cdot (-4) = -8x.
    • Then 3⋅x=3x3 \cdot x = 3x and 3⋅(−4)=−123 \cdot (-4) = -12.
    • The four products are 2x22x^2, −8x-8x, 3x3x and −12-12.
    • The minus sign changes both signs in the second parentheses: −(x2−5)-(x^2 - 5) is −x2+5-x^2 + 5.

  2. Step 2: Combine like terms

    • The x2x^2 terms: 2x2−x2=x22x^2 - x^2 = x^2.
    • The xx terms: −8x+3x=−5x-8x + 3x = -5x.
    • The numbers: −12+5=−7-12 + 5 = -7.
    • The result is x2−5x−7x^2 - 5x - 7.
The correct answer choice is A: x2−5x−7x^2 - 5x - 7.

The same example in Desmos

  1. new graph

    Keys: y=(2x+3)(x-4)-(x^2 → -5)

    The given expression, as a curve.

  2. Keys: y=x^2 → -5x-7

    Choice A lies exactly on top of it (dashed in the picture).

Type
Desmos expression list: y=(2x+3)(x-4)-(x^2-5); y=x^2-5x-7
Look
Desmos graph of y=(2x+3)(x-4)-(x^2-5); y=x^2-5x-7

Made with Desmos

Open Desmos

Your turn

The expression 9x2−259x^2 - 25 can be written as (3x+c)(3x−c)(3x + c)(3x - c), where cc is a positive constant. What is the value of cc?

Show the answerHide the answer

Answer

The correct answer is 55.

  1. Step 1: Spot the special product

    • 9x2=(3x)29x^2 = (3x)^2 and 25=5225 = 5^2.
    • So 9x2−259x^2 - 25 is a difference of squares.

  2. Step 2: Factor the expression

    • a2−b2a^2 - b^2 =(a+b)(a−b)= (a + b)(a - b), with a=3xa = 3x and b=5b = 5.
    • 9x2−259x^2 - 25 =(3x+5)(3x−5)= (3x + 5)(3x - 5). So c=5c = 5.
    • Check with FOIL: 3x⋅3x=9x23x \cdot 3x = 9x^2 and 5⋅(−5)=−255 \cdot (-5) = -25.
    • The two middle products are −15x-15x and 15x15x. They add to 00.
    • So the product is 9x2−259x^2 - 25. It matches.

Traps

  • Not changing the second sign. −(x2−5)-(x^2 - 5) is −x2+5-x^2 + 5, not −x2−5-x^2 - 5. With −5-5 the number is −12−5=−17-12 - 5 = -17: choice B, a wrong answer.
  • Adding the x2x^2 instead of subtracting it. 2x2+x22x^2 + x^2 =3x2= 3x^2 gives choice D, a wrong answer. The minus sign applies to x2x^2 too.
Practice this lesson64 questions

Radical equations and extraneous solutions

Key points

  • Get the square root alone on one side, then square both sides.
  • Squaring can create an answer that does not work: an extraneous solution.
  • Check every answer in the original equation.
  • A square root is never negative. If it would have to equal a negative number, reject that answer.
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See it solved

Example

2x+3=x\sqrt{2x + 3} = x

What is the solution to the given equation?

Worked by hand

The plan

  1. Square both sides
  2. Solve the equation
  3. Check for an extraneous solution

Desmos is quicker here, and no check is needed: type the equation as given. An extraneous solution never appears.

  1. Step 1: Square both sides

    • The square root is already alone.
    • Squaring removes the root: 2x+3=x22x + 3 = x^2.

  2. Step 2: Solve the equation

    • Subtract 2x2x and 33 from both sides: 0=x2−2x−30 = x^2 - 2x - 3.
    • Two numbers that multiply to −3-3 and add to −2-2: they are −3-3 and 11.
    • (x−3)(x+1)(x - 3)(x + 1) =0= 0, so x=3x = 3 or x=−1x = -1.

  3. Step 3: Check for an extraneous solution

    • x=3x = 3: 2⋅3+3\sqrt{2 \cdot 3 + 3} =9= \sqrt{9} =3= 3. The right side is 33. It works.
    • x=−1x = -1: 2⋅(−1)+3\sqrt{2 \cdot (-1) + 3} =1= \sqrt{1} =1= 1. The right side is −1-1.
    • They are different, so −1-1 is extraneous. The only solution is 33.
The correct answer is 33.

The same example in Desmos

  1. new graph

    Keys: sqrt 2x+3 → =x

    One vertical line.

  2. read where the line meets the x-axis; this line has no point to click

    x=3x = 3. There is no line at x=−1x = -1: Desmos does not draw extraneous solutions.

Type
Desmos expression list: sqrt{2x+3}=x
Look
Desmos graph of sqrt{2x+3}=x, with (3, 0) labelled

Made with Desmos

Open Desmos

Your turn

x−1+3=7\sqrt{x - 1} + 3 = 7

What is the solution to the given equation?

Show the answerHide the answer

Answer

The correct answer is 1717.

  1. Step 1: Square both sides

    • First get the root alone. Subtract 33 from both sides: x−1=4\sqrt{x - 1} = 4.
    • Square both sides: x−1=42=16x - 1 = 4^2 = 16.

  2. Step 2: Solve the equation

    • Add 11 to both sides: x=16+1=17x = 16 + 1 = 17.

  3. Step 3: Check for an extraneous solution

    • 17−1+3\sqrt{17 - 1} + 3 =16+3= \sqrt{16} + 3 =4+3= 4 + 3 =7= 7. It works.

Traps

  • Keeping the extraneous answer. x=−1x = -1 solves 2x+3=x22x + 3 = x^2, but not the original equation: 1\sqrt{1} is 11, not −1-1. Answering −1-1 is wrong.
  • Squaring before the root is alone. In x−1+3=7\sqrt{x - 1} + 3 = 7, squaring first leaves a square root in the middle term and a harder equation. Subtract 33 first.
Practice this lesson36 questions
Typing tips for Desmos
  • In the keys lines, → means the right-arrow key.
  • Typing / starts a fraction, and everything after it goes into the denominator until you press →. So 42/120*100 typed straight through becomes 42 ÷ (120 × 100). Type 42/120 → *100 instead.
  • Typing ^ or sqrt works the same way: press → to come back down from an exponent or out of a square root before typing the rest.
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