1.15 Chapter Review

1.15.1 Chapter summary

Real-number properties, fractions, sets, exponents, and roots establish the language of algebra. Polynomial operations and factoring reveal structure, while polynomial division and zero theorems support further analysis. Equations determine values, lines model constant rates, and inequalities identify ranges. Rational and radical expressions require explicit domain checks. Every applied result should be checked algebraically, dimensionally, and physically.

1.15.2 Common mistakes

  • Omitting units or combining incompatible units.
  • Confusing display-math delimiters with interval brackets.
  • Cancelling terms rather than common factors.
  • Dropping domain restrictions after simplification.
  • Forgetting to reverse an inequality after division by a negative value.
  • Accepting an extraneous or physically impossible answer.

1.15.3 Exercises

  1. Write \(-4<x\le2.5\) in interval notation.
Check Your Work \((-4,2.5]\).

Check Your Work. \((-4,2.5]\).

  1. Find the distance between elevations \(-1.8\) m and 4.7 m.
Check Your Work \(|4.7-(-1.8)|=6.5\) m.

Check Your Work. 6.5 m.

  1. Simplify \(x^3x^{-2}\) and state the original restriction.
Check Your Work \(x\), with \(x\ne0\).

Check Your Work. \(x\), with \(x\ne0\).

  1. Factor \(4x^2-8x-21\).
Check Your Work \((2x-7)(2x+3)\).

Check Your Work. \((2x-7)(2x+3)\).

  1. Find the remainder when \(2x^3-7x^2+5\) is divided by \(x-3\).
Check Your Work \(P(3)=54-63+5=-4\).

Check Your Work. \(-4\).

  1. Find the line through \((2,1)\) and \((6,9)\).
Check Your Work \(m=2\), so \(y=2x-3\).

Check Your Work. \(y=2x-3\).

  1. Simplify \((x^2-9)/(x-3)\) and state its restriction.
Check Your Work \(x+3\), with \(x\ne3\).

Check Your Work. \(x+3\), with \(x\ne3\).

  1. Solve \(3x^2-10x=8\).
Check Your Work \(x=-2/3\) or \(x=4\).

Check Your Work. \(x=-2/3\) or \(x=4\).

  1. Solve \(4<5x-6\le19\).
Check Your Work \(2<x\le5\), or \((2,5]\).

Check Your Work. \(2<x\le5\), or \((2,5]\).

  1. A basin has width \(w\), length \(w+5\), and area 84 m². Find its dimensions.
Check Your Work \(w^2+5w-84=(w+12)(w-7)=0\). Reject \(w=-12\). The dimensions are 7 m by 12 m.

Check Your Work. 7 m by 12 m.

1.15.4 Questions for discussion

  1. Why should domain restrictions be stated before simplifying a rational expression?

  2. When can a negative solution be meaningful in a water-engineering model?

  3. How can units reveal an algebra or formula-entry error?