Keep every absolute value in a logarithmic inequality

Sketch the graphs of y=ln⁡(1−∣x∣)y=\ln(1-|x|) and y=−∣x−1∣y=-|x-1| on the same diagram, showing clearly the equations of any asymptotes and any intersections with the axes.

Hence, solve the inequality ln⁡(1−∣x∣)+∣x−1∣≤0\ln(1-|x|)+|x-1|\le0.

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Solution

Sketch

Curve sketching

Sketch the graphs of y=ln⁡(1−∣x∣)y=\ln(1-|x|) and y=−∣x−1∣y=-|x-1| on the same diagram, showing clearly the equations of any asymptotes and any intersections with the axes.

Explore this idea

Hint

  1. Where is ln⁡(1−∣x∣)\ln(1-|x|) defined? Look for the largest value of 1−∣x∣1-|x|.

  2. −∣x−1∣-|x-1| is a V flipped upside down. Where is its vertex?

  3. Mark the axis intercepts and the asymptotes of each.

Q10 logarithm and absolute line
Q10 logarithm and absolute lineScroll across to see the full diagram.

Answer · Sketch

ln⁡(1−∣x∣)\ln(1-|x|): maximum (0,0)(0,0), asymptotes x=±1x=\pm1. −∣x−1∣-|x-1|: vertex (1,0)(1,0), intercept (0,−1)(0,-1).

Explore the idea

A logarithm and a flipped V

Switch on each feature. The curves meet twice inside the strip −1 < x < 1.

Switch on a feature to add it to the sketch and see how it is found. A full-mark sketch shows all of them.

  • Y = ln(1 − |x|)
  • Y = −|x − 1|
  • Asymptotes x = ±1
  • Intersections of the two curves
Ideas to try
  1. Which curve is higher in the middle, near x=0x=0?
  2. Which side of x=−0.841x=-0.841 has the logarithm below the V?
Solve

Inequalities from graphs

Hence solve the inequality ln⁡(1−∣x∣)+∣x−1∣≤0\ln(1-|x|)+|x-1|\le0.

Hint

  1. Rearrange so the two sketched curves appear.

  2. ≤0\le0 means the first curve is below or on the second. Read off the intervals.

  3. Remember the domain of the logarithm.

Answer · Solve

−1<x≤−0.841406…-1<x\le-0.841406\ldots or 0.432857…≤x<10.432857\ldots\le x<1.

Cutoffs solve the two curve intersections; ±1 are excluded.