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Absolute value

Split into two cases and check each.

|u| = 5 means u is 5 units from zero in either direction, so u = 5 or u = −5. Every absolute-value equation splits into those two cases; solve each and keep the solutions consistent with the case they came from.

Օրինակ: |x - 2| = 5

Algebra · Absolute-Value

Solve Abs(x - 2) = 5

\left|{x - 2}\right| = 5

Քայլ առ քայլ

  1. \left|{x - 2}\right| = 5

    Start from the equation as given.

  2. \left|{x - 2}\right| - 5 = 0

    Split into the two cases inside the absolute value: positive and negative.

  3. x - 7 = 0 \Rightarrow x = 7

    Case x - 2 non-negative: replace |x - 2| by (x - 2) and solve; keep only solutions consistent with that case.

  4. - x - 3 = 0 \Rightarrow x = -3

    Case x - 2 negative: replace |x - 2| by −(x - 2) and solve; keep only solutions consistent with that case.

Առաջարկել պատասխանը
x = 7 ,\; x = -3

Symbols used here

\neg,\ \wedge,\ \vee,\ \Rightarrow,\ \Leftrightarrow
not, and, or, implies, iff
Logical connectives.
|x|
absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\pm
plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.
\neq
not equal
The two sides are different.
\sqrt{x},\ \sqrt[n]{x}
square root, n-th root
The non-negative number whose square (n-th power) is x.
i
imaginary unit
i² = −1.
\log_b x,\ \ln x
logarithm, natural log
The exponent b must be raised to for x; ln uses base e.
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Absolute value

  1. Start from the equation as given.
  2. Split into the two cases inside the absolute value: positive and negative.
  3. Case x - 2 non-negative: replace |x - 2| by (x - 2) and solve; keep only solutions consistent with that case.
  4. Case x - 2 negative: replace |x - 2| by −(x - 2) and solve; keep only solutions consistent with that case.

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

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