Algebra II End-of-Course Exam Content Standards—Core: Equations and Inequalities (Priority: 20%)

Successful students will be able to solve and graph the solution sets of equations and inequalities and systems of linear equations and inequalities. The types of equations are to include linear, linear absolute value, quadratic, exponential, rational, radical, and higher order polynomials; the types of inequalities are to include linear and quadratic. There are a variety of types of test items including some that cut across the objectives in this standard and require students to make connections and, where appropriate, solve contextual problems. In contextual problems students will be required to graph and interpret their solutions in terms of the context. (Contextual test items will be limited to inequalities, systems of equations and inequalities, and those equations that do not represent a function.)

E1. Linear equations and inequalities

a. Solve equations and inequalities involving the absolute value of a linear expression.

Example: |x - 6| ≤ 8 ↔ (x - 6) ≤ 8 and -(x - 6) ≤ 8 ↔ x ≤ 14 and x ≥ -2 ↔ -2 ≤ x ≤ 14

b. Express and solve systems of linear equations in three variables with and without the use of technology.

• Systems in three variables should be limited to those with integer solutions and small integral coefficients.

  Example: Solve the system

  2x+z=11 x+2y=7 2x-4y+z=3

  OR

  x-y+z=6 2x+2y-z=1 3x+4y+3z=4

c. Solve systems of linear inequalities in two variables and graph the solution set.

Example: Graph the solution set of the following system.
3x - 2y < 7, x ≥ 0, y ≤ 0

• Recognize that the graphic solution of a linear inequality is either an open or closed half plane.

  Example: Graphs (a) and (b) illustrate that the graph of a linear inequality is a half plane. Graph (c) illustrates a solution to the question: What is the set of points (x, y) that satisfies both 5x - y ≥ 3 and 2x - 4y < 1?

  a.

  b.

  c.

d. Recognize and solve problems that can be represented by single variable linear equations or inequalities or systems of linear equations or inequalities involving two or more variables. Interpret the solution(s) in terms of the context of the problem.

• This includes using and interpreting appropriate units of measurement and precision for the given application.
• Common problems are those that involve, time/rate/distance, percentage increase or decrease, ratio and proportion; mixture problems and break-even problems.

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E2. Nonlinear equations and inequalities

a. Solve single-variable quadratic, exponential, rational, radical, and factorable higher-order polynomial equations over the set of real numbers, including quadratic equations involving absolute value.

• Use information gathered from a polynomial equation to determine the number and nature of its solutions:
  • The number of roots, real and complex, counted by multiplicity, for a polynomial is equal to its degree.
  • Some polynomial equations, including quadratics, may have no real solutions.
  • All complex solutions come in pairs (conjugates) making it necessary that all odd polynomial functions have at least one real solution.
• Explain the relationship between the number of real (and complex) solutions and the graph of a polynomial equation.
• Solve equations and inequalities numerically, graphically, and algebraically, with and without the use of technology, making connections between solution strategies.
• Solve power equations with integer exponents, axⁿ = b algebraically, graphically, and using technology.

  Power equations can represent area or volume; polynomial equations can represent projectile height, profit, or revenue.

• Solve radical equations numerically, algebraically, and graphically, with and without the use of technology.
• Use the factored form of a polynomial to determine its real roots.
• Consider domain restrictions (asymptotes or undefined values) when finding solutions of rational and radical equations.
• Know which operations on an equation may produce an equation with the same solutions, and which produce an equation with fewer or more solutions (lost or extraneous solutions).

b. Solve single variable quadratic equations and inequalities over the complex numbers; graph real solution sets on a number line.

• Solve quadratic equations and inequalities using factoring, completing the square, and the quadratic formula.
• Use a calculator to approximate the solutions of a quadratic equation related to an inequality and as an aid in graphing.
• Select and explain a method of solution (e.g., exact vs. approximate) that is effective and appropriate for a given problem.
• Determine a single variable quadratic equation given its solutions.
• Recognize that complex solutions come in conjugate pairs of the form a + bi and a – bi.

Task related to this benchmark: Leo's Painting

c. Use the discriminant, D = b² - 4ac, to determine the nature of the solutions of the equation ax² + bx + c = 0.

• Describe how the discriminant, D = b² - 4ac, indicates the nature of the solutions of the equation ax² + bx + c = 0.

  The solutions are real and distinct if D > 0; real and equal if D = 0; and complex if D < 0.

d. Graph the solution set of a two-variable quadratic inequality in the coordinate plane.

• Students might also be asked to write or identify the quadratic inequality from the graph.

e. Rewrite nonlinear equations and inequalities to express them in multiple forms in order to facilitate finding a solution set or to extract information about the relationships or graphs indicated.

• Inequalities and equations that do not represent functions may be included. (e.g. horizontal parabolas)

Example: Explain how you know how to find the x-intercept for the equation x = y² - 6y + 8.

Example: For y + 1 ≥(x - 3)² find the vertex and sketch a graph.

• Inequalities and equations in one variable will be included.

Example: x² + 8x - 7 - 0 ↔ (x + 7)(x + 1) - 0 ↔ x = -7 or x = -1

• Rewriting or solving a formula in several variables for one variable in terms of the others will be included.

Example: Solve V = πr²h for r or h.

Example: Solve A = πr² + 2πrh for r or h.

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