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Absolute Value (Medium)

AMC 10/12 mid (≈Q9–16) — multi-step.

EasyMediumHardAIME / USAJMO / AMO

Absolute Value: core idea

Equations.

Example problem: hint and answer

Solve ∣x−4∣=7|x-4|=7 for the smaller xx.

Hint: Split an absolute-value equation into its two sign cases. Answer: -3.

Practice (9 problems — type your answers)

  • 1. How many solutions has ∣x+2∣=5|x+2|=5?

  • 2. Solve ∣2x∣=10|2x|=10 for positive xx.

  • 3. Find xx if ∣x−3∣+2=7|x-3|+2=7 and x>3x>3.

  • 4. Solve ∣x+1∣=∣x−5∣|x+1|=|x-5|.

  • 5. Find the minimum of ∣x−4∣|x-4|.

  • 6. Solve ∣3x−6∣=9|3x-6|=9 for the larger xx.

  • 7. How many integers satisfy ∣x∣<4|x|<4?

  • 8. How many integers satisfy ∣x−2∣≤3|x-2|\le3?

  • 9. Find ∣−3∣⋅∣−4∣|{-3}|\cdot|{-4}|.

Absolute Value Practice Problems — Medium