AlgebraStructure & Method · Book 1All Chapters

Problem Solving

4-10Problems Without Solutions

Overview

Sometimes the equation simplifies to a false statement (no solution) or an identity (all real numbers).

Key ideas

  • If you reach 3 = 7, there is no solution.
  • If you reach 0 = 0, every real number works.
  • Always check at the end.

Worked examples

Example 1

Problem. Solve 2(x + 3) = 2x + 5.

Solution. 2x + 6 = 2x + 5 → 6 = 5 (false). No solution.

Example 2

Problem. Solve 3(x − 1) = 3x − 3.

Solution. 3x − 3 = 3x − 3 → identity. All real numbers.

Practice

Try each one. Click Show answer when ready.

  1. 1.

    Solve 4x + 1 = 4x − 2.

  2. 2.

    Solve 5(x + 2) = 5x + 10.

  3. 3.

    Solve 2x + 3 = 2x.

Challenge problems

A little tougher — great for test prep. Click Show answer when ready.

  1. 1.

    Show that 2x + 3 = 2x + 5 has no solution.

  2. 2.

    How many solutions does 3(x + 2) = 3x + 6 have? Explain.