Algebra Midterm #2 Review

• §9.1: Inequalities and Domain
  • Meaning of <, >, ≤, ≥
  • Solving inequalities algebraically or graphically
  • Remember: multiplying or dividing by a negative reverses the inequality!
  • Writing solution as a number line graph, in interval notation, in set-builder notation
  • Finding domain of square root functions (set radicand ≥ 0 and solve)


• §9.2: Intersections, Unions, and Compound Inequalities
  • Set theory operations:
    • Intersection: A∩B = {all things that are in A and in B}
    • Union: A∪B = {all things that are in A or in B}
    • Note that A and B can be intervals
  • Compound Inequalities: two or more inequalities combined with "and" or "or"
    • Mathematically, "and" means that all of the statements must be true
    • Mathematically, "or" means that at least one of the statements must be true
    • Be careful which one to use in the solution!


• §9.3: Absolute Value Equations and Inequalities
  • Geometrically, |A| means "distance from A to 0"
  • Algebraically, |A| means "A" if A ≥ 0, or "−A" if A < 0
  • This means that if |A| = B, then A might be B or A might be −B.
    • So write "A = B or A = −B" and solve both equations.
    • Note that if B is negative, there is no solution!
    • You may need to isolate the absolute value first.
  • To solve an inequality with absolute value, first treat it as an equation.
    • Solving this equation will tell you the boundary points.
    • Mark boundary points on a number line and test a sample point in each interval.
    • This will determine where the original inequality is true and where it's false.
    • Write solution set in interval notation or as a compound inequality.


• §9.4: Inequalities in Two Variables (typically x & y)
  • Again, treat it as an equation to graph the boundary line
    • Use a solid line for ≤ (boundary included) or dashed for < (boundary excluded)
  • And again, test a sample point not on the line (origin is usually easiest)
    • If that point makes the inqeuality true, shade that side; otherwise, try another point
  • Systems of inequalities: shade only the region that makes ALL of the inequalities true
    • Try graphing one at a time, and use small arrows to denote "shaded" side
    • This is useful for describing a region of the plane
    • Each inequality can be thought of as "slicing" what's left of the plane

• §10.1: Radical Expressions & Functions
  • Terminology: radical, radicand, index, principal square root
  • Difference between odd and even roots when dealing with negatives


• §10.2: Rational Numbers as Exponents
  • Review the Laws of Exponents! (page 645)
  • Fractions as exponents: A^1/n = ⁿ√(A); more generally A^m/n = ⁿ√(A^m) = ⁿ√(A)^m
  • A negative exponent means taking the reciprocal of the base; A⁻ⁿ = 1/Aⁿ
  • Sometimes converting all radicals to exponents can be helpful for simplifying.


• §10.3: Multiplying Radical Expressions
  • Since radicals are exponents, they distribute over multiplication: ⁿ√(A·B) = ⁿ√(A) · ⁿ√(B).
  • To simplify, factor out any perfect n^th powers and split the radical.


• §10.4: Dividing Radical Expressions
  • Similarly, radicals distribute over division: ⁿ√(A/B) = ⁿ√(A) / ⁿ√(B)
  • Sometimes combining into one radical first, reducing, and THEN simplifying works well.
  • Don't leave any fractions in radicals, and don't leave any radicals in the denominator!
  • Rationalizing the denominator:
    • multiply top and bottom by whatever is needed to cancel the denominator's radical


• §10.5: Expressions Containing Several Radical Terms
  • In sums, identical radicals can be factored out, or combined as like terms (same thing)
  • Sometimes "like terms" may appear after all radicals are fully simplified


• §10.6: Solving Radical Equations
  • Isolate the radical and raise both sides to a power to cancel out the root.
  • If there are two radicals, isolate one and raise to a power. Repeat for the other.
  • Once the radicals are gone, solve what's left (typically linear or quadratic).
  • Always check for extraneous solutions, especially when dealing with even roots!


• §10.7: Geometrical Applications
  • Pythagorean Theorem: (leg₁)² + (leg₂)² = (hypotenuse)²
    • Substitute in the measurements you know and solve for the one you don't.
  • "Special Triangles": 45°-45°-90° (isosceles) and 30°-60°-90° (half of an equilateral)
    • You can always re-work these from scratch if you forget the proportions.
  • Distance Formula: distance = √( (Δx)² + (Δy)² ) ... really the Pythagorean Theorem!

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