When B is zero or negative

Every length sits at zero or above. A negative B lies below the whole ruler. Equality and a smaller distance cannot reach it. A greater distance is larger than it even at zero.

• |A| < B or |A| ≤ B with B < 0: no solution.
• |A| > B or |A| ≥ B with B < 0: all allowed inputs.
• |A| = B with B < 0 also has no solution, because it requires a negative distance.

A zero length leash keeps you at its post. A requirement for positive distance sends you anywhere except the post. At a zero B, inspect whether equality is required, forbidden, or allowed along with every positive distance.

• |A| = 0 and |A| ≤ 0 both mean A = 0.
• |A| > 0 means A ≠ 0. The symbol ≠ means 'not equal to'.
• |A| < 0 is impossible. |A| ≥ 0 accepts every input where A is defined.

If you flatten both ramps completely, they stop making a V. In a|x − h| + k, choosing a = 0 removes the distance part. Every input then gives the same height k. This is a constant function, a flat horizontal line. The choice of h cannot create a corner because zero times any distance is still zero.

• If a = 0, f(x) = 0|x − h| + k = k, a constant function with no unique corner.
• Domain: (−∞, ∞). Range: {k}, a set containing the one output k.
• If k ≠ 0 there are no zeros. If k = 0, every real input is a zero.

Pack for this section the way you pack for a short trip. Keep the few ideas you need constantly in your pocket. Learn how to rebuild the details from those ideas. Keep a compact reference for checking notation during practice. The three lists below separate those jobs. For a closed-book exam, use the reference while studying, then cover it and explain each line. You do not need to memorize every worked answer or every transformed V. You can rebuild them from distance, the corner, and the two straight arms.

• know cold
  Absolute value is nonnegative distance. Memory device: distance has size, direction has sign. This explains |−6| = 6 and |0| = 0.
  Isolate before splitting. Memory device: alone, then two roads. This keeps outside arithmetic from entering the two cases.
  A positive number B gives two inside values, a number B equal to 0 gives one, and a negative number B gives none. Memory device: two, one, none. This follows from the two sides of home.
  Less than means between with and; greater than means outside with or, for a positive limit. Memory device: less thAND, greatOR. This remembers whether one region or two regions work.
  Multiplying or dividing an inequality by a negative reverses its sign. Memory device: a negative turns the street around. This remembers that left and right trade places.
  A bracket includes an endpoint; a parenthesis excludes it. Memory device: a bracket grabs. This remembers whether equality is allowed.
• understand, then rebuild it when needed
  Rebuild |x| = x for x ≥ 0 and |x| = −x for x < 0 from distance, because the negative address needs its direction removed.
  Rebuild the corner and arm slopes of a|x − h| + k from x = h and a one-unit step, because zero inside the bars marks the turn.
  Rebuild shifts, reflection, horizontal scaling, domain, and range from the parent V, because these describe movements or changes in its two arms.
  Rebuild an equation from a corner and a second point by substitution, because the point supplies the missing stretch factor a.
  Rebuild zeros and positive or negative regions from boundaries and a graph or test point, because outputs can change sign only through a zero.
  Do not memorize each finished graph or individual answer. These can be rebuilt from the picture and a few arithmetic steps.
• put on the cheat sheet
  Keep the V formula, the equation cases, the between and outside cases, and endpoint notation together, because these are the short lines you compare during practice.
  Keep the zero and negative limit cases beside them, because the positive-limit shortcuts do not cover those cases.
  Keep slope, percent tolerance, and the reciprocal of a negative fraction visible, because arithmetic slips can spoil an otherwise correct solution.
  Use the printable reference below as one page. For a closed-book exam, practice rebuilding it from memory rather than bringing it.
• A small set of pictures explains the whole reference. Distance explains the cases. A corner and one step explain the V. Number-line order explains inequalities and endpoints. The compact sheet records the results so you can compare your reasoning during practice.
  
  One-page cheat sheet: absolute value functions
  Meaning: |x| is distance from 0. |x − c| is distance from c. Distance is always ≥ 0.
  Piecewise: |x| = x for x ≥ 0; |x| = −x for x < 0.
  Graph: f(x) = a|x − h| + k, a ≠ 0. Corner (h, k). Right arm slope a; left arm slope −a. Up if a > 0; down if a < 0. Domain (−∞, ∞). Range [k, ∞) if a > 0; (−∞, k] if a < 0.
  Scaling: |b(x − h)| = |b|·|x − h|. A nonzero inside factor b changes widths by 1 ÷ |b|.
  Equation: isolate |A| = B. For B > 0, A = B or A = −B. For B = 0, A = 0. For B < 0, no solution. Check the original.
  For B > 0: |A| < B means −B < A < B. |A| ≤ B means −B ≤ A ≤ B.
  For B > 0: |A| > B means A < −B or A > B. |A| ≥ B means A ≤ −B or A ≥ B.
  For B < 0: |A| < B and |A| ≤ B have no solution; |A| > B and |A| ≥ B hold for every allowed input.
  For B = 0: |A| < 0 has no solution; |A| ≤ 0 means A = 0; |A| > 0 means A ≠ 0; |A| ≥ 0 holds for every allowed input.
  Inequalities: add or subtract everywhere. Multiply or divide by a negative and reverse every sign.
  Intervals: (p, q) excludes both ends; [p, q] includes both. Mix symbols as needed. Infinity always takes a parenthesis. ∪ means union, combining allowed regions.
  Intercepts: set output y = 0 for horizontal intercepts; set input x = 0 for the vertical intercept.
  Slope = change in ychange in x$\frac{changeiny}{changeinx}$. Percent amount = percent100$\frac{percent}{100}$ × center.
  Arithmetic: divide by −12$\frac{1}{2}$ by multiplying by −2. Fractions must have matching bottoms before addition or subtraction.