Algebra · Lesson 5

Absolute value and rational equations

Absolute value is distance from 0. ∣A∣=b|A| = b with b>0b > 0 means A=bA = b or A=−bA = -b, so you solve two equations. For ∣2x−5∣=9|2x - 5| = 9, solve 2x−5=92x - 5 = 9 and 2x−5=−92x - 5 = -9 to get x=7x = 7 or x=−2x = -2. If bb is negative, as in ∣x∣=−2|x| = -2, there is no solution, because a distance is never negative. Isolate the absolute value first: ∣x−4∣+9=3|x - 4| + 9 = 3 becomes ∣x−4∣=−6|x - 4| = -6, which has no solution.

Absolute value inequalities. ∣A∣<b|A| < b means −b<A<b-b < A < b: one interval between two numbers. ∣A∣≥b|A| \ge b means A≤−bA \le -b or A≥bA \ge b: two rays pointing away from each other. So ∣x−3∣<4|x - 3| < 4 gives −1<x<7-1 < x < 7, and ∣x−3∣≥4|x - 3| \ge 4 gives x≤−1x \le -1 or x≥7x \ge 7.

Rational equations. Multiply every term by the common denominator to clear the fractions, then solve the equation that is left. Before you answer, list the values that make any denominator 0. A candidate on that list is extraneous: it solves the cleared equation but not the original one, so you throw it out.

Worked example Easy

What are all the solutions of the equation ∣2x−5∣=9|2x - 5| = 9?

  1. x=7x = 7 only
  2. x=−7x = -7 or x=7x = 7
  3. x=−2x = -2 or x=7x = 7Answer
  4. x=2x = 2 or x=7x = 7

How to solve it

  1. Split into two cases: 2x−5=92x - 5 = 9 or 2x−5=−92x - 5 = -9.
  2. First case: 2x=142x = 14, so x=7x = 7.
  3. Second case: 2x=−42x = -4, so x=−2x = -2.
  4. Check: ∣14−5∣=9|14 - 5| = 9 and ∣−4−5∣=∣−9∣=9|-4 - 5| = |-9| = 9.

Why each choice is right or wrong

  • A. Incorrect. x=7x = 7 only solves just the positive case 2x−5=92x - 5 = 9. The expression inside can also equal −9-9, which gives a second solution.
  • B. Incorrect. x=−7x = -7 or x=7x = 7 comes from subtracting 5 in the second case: 2x=−9−5=−142x = -9 - 5 = -14. To undo −5-5, add 5: 2x=−9+5=−42x = -9 + 5 = -4.
  • C. Correct. 2x−5=92x - 5 = 9 gives x=7x = 7, and 2x−5=−92x - 5 = -9 gives x=−2x = -2. Both check in the original equation.
  • D. Incorrect. x=2x = 2 or x=7x = 7 writes the second case as 2x+5=92x + 5 = 9, negating only the 5. The second case is 2x−5=−92x - 5 = -9, which gives x=−2x = -2.

Worked example Medium

What is the solution set of the equation xx−3+10x=3x−3\frac{x}{x - 3} + \frac{10}{x} = \frac{3}{x - 3}?

  1. x=−10x = -10 onlyAnswer
  2. x=3x = 3 only
  3. x=−10x = -10 and x=3x = 3
  4. No real solution

How to solve it

  1. The denominators are 0 at x=3x = 3 and x=0x = 0, so neither can be a solution.
  2. Multiply every term by x(x−3)x(x - 3): x2+10(x−3)=3xx^2 + 10(x - 3) = 3x.
  3. Simplify: x2+7x−30=0x^2 + 7x - 30 = 0, which factors as (x+10)(x−3)=0(x + 10)(x - 3) = 0.
  4. Reject x=3x = 3, because it makes x−3=0x - 3 = 0. Only x=−10x = -10 remains.

Why each choice is right or wrong

  • A. Correct. Clearing denominators gives (x+10)(x−3)=0(x + 10)(x - 3) = 0. The value 3 makes a denominator 0, so only x=−10x = -10 solves the original equation. Check: −10−13+10−10=−313\frac{-10}{-13} + \frac{10}{-10} = -\frac{3}{13} and 3−13=−313\frac{3}{-13} = -\frac{3}{13}.
  • B. Incorrect. x=3x = 3 only keeps the wrong candidate. At x=3x = 3 the denominator x−3x - 3 is 0, so 3 is extraneous, while −10-10 works.
  • C. Incorrect. x=−10x = -10 and x=3x = 3 are the solutions of the cleared equation, but 3 makes xx−3\frac{x}{x - 3} undefined, so it must be thrown out.
  • D. Incorrect. No real solution rejects both candidates. Only 3 makes a denominator 0; x=−10x = -10 checks out in the original equation.

Trap Solving only one case

An absolute value equation like ∣2x−5∣=9|2x - 5| = 9 almost always has two solutions. A common distractor shows only the one from the positive case. Write both cases before you solve, and in the negative case put the whole right side in negative form: 2x−5=−92x - 5 = -9, not 2x+5=92x + 5 = 9.

Trap Keeping a value that breaks a denominator

Clearing fractions can produce a value that makes an original denominator 0. Write down the forbidden values before you multiply, and cross them off at the end. If every candidate is forbidden, the equation has no solution.

Trap Interval or two rays

Less than gives one interval between two numbers: ∣x−3∣<4|x - 3| < 4 is −1<x<7-1 < x < 7. Greater than gives two rays: ∣x−3∣>4|x - 3| > 4 is x<−1x < -1 or x>7x > 7. Test a value such as x=3x = 3 to confirm which picture is right.

Calculator When a calculator isn’t faster

Hand work is usually quicker here: two short cases for an absolute value, or one multiplication to clear denominators. Your calculator is best as a check. Plug each candidate into the original equation; an extraneous value gives an error because it divides by 0.

With the built-in Desmos calculator if you test online, graph y = abs(2x - 5) and y = 9 to see both crossings, or graph y = abs(x + 2) and y = 5 to see the interval where the V sits below the line.