Equations and Inequalities

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Questions and Answers

What makes an equation consist of?

  • Only one mathematical expression.
  • Expressions on both sides separated by an inequality symbol.
  • Two variables separated by an inequality symbol.
  • Expressions on both sides separated by an equal sign. (correct)

Which method is not commonly used to solve both equations and inequalities?

  • Graphing (correct)
  • Multiplication/Division
  • Addition/Subtraction
  • Substitution

Which is a characteristic of a quadratic equation?

  • The equation has a variable with a coefficient of 1.
  • The highest power of the variable is 2. (correct)
  • The highest power of the variable is 1.
  • The equation has no variable.

What is typically the first step when using addition or subtraction to solve an equation?

<p>Isolate the constants on one side. (B)</p> Signup and view all the answers

What is true about linear inequalities?

<p>The highest power of the variable is 1. (C)</p> Signup and view all the answers

Which type of equation is represented by $2x + 3 = 5$?

<p>Linear Equation (B)</p> Signup and view all the answers

Which of the following changes the direction of the inequality symbol?

<p>Multiplying or dividing both sides by a negative value. (D)</p> Signup and view all the answers

Which variable value makes the quadratic equation $x^2 + 4x + 4 = 0$ true?

<p>x = -2 (B)</p> Signup and view all the answers

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Study Notes

Equations

  • An equation is a statement that says two mathematical expressions are equal.
  • It consists of two parts: the left-hand side (LHS) and the right-hand side (RHS), separated by an equal sign (=).
  • The goal is to find the value of the variable(s) that makes the equation true.

Types of Equations:

  • Simple Equations: Equations in which the variable has a coefficient of 1, e.g., 2x = 5.
  • Linear Equations: Equations in which the highest power of the variable is 1, e.g., 2x + 3 = 5.
  • Quadratic Equations: Equations in which the highest power of the variable is 2, e.g., x^2 + 4x + 4 = 0.

Methods for Solving Equations:

  • Addition/Subtraction: Add or subtract the same value to both sides of the equation to isolate the variable.
  • Multiplication/Division: Multiply or divide both sides of the equation by the same non-zero value to isolate the variable.
  • Substitution: Substitute a value or expression into the equation to solve for the variable.

Inequalities

  • An inequality is a statement that says two mathematical expressions are not equal.
  • It consists of two parts: the left-hand side (LHS) and the right-hand side (RHS), separated by an inequality symbol (<, >, ≤, ≥).
  • The goal is to find the range of values that makes the inequality true.

Types of Inequalities:

  • Linear Inequalities: Inequalities in which the highest power of the variable is 1, e.g., 2x + 3 > 5.
  • Quadratic Inequalities: Inequalities in which the highest power of the variable is 2, e.g., x^2 + 4x + 4 > 0.

Methods for Solving Inequalities:

  • Addition/Subtraction: Add or subtract the same value to both sides of the inequality to isolate the variable.
  • Multiplication/Division: Multiply or divide both sides of the inequality by the same non-zero value, but be careful with the direction of the inequality symbol.
  • Graphing: Graph the related function and find the range of values that makes the inequality true.

Note: When solving inequalities, the direction of the inequality symbol changes when multiplying or dividing by a negative value.

Equations

  • An equation is a statement that says two mathematical expressions are equal.
  • It consists of two parts: the left-hand side (LHS) and the right-hand side (RHS), separated by an equal sign (=).
  • The goal is to find the value of the variable(s) that makes the equation true.

Types of Equations

  • Simple Equations: equations in which the variable has a coefficient of 1, e.g., 2x = 5.
  • Linear Equations: equations in which the highest power of the variable is 1, e.g., 2x + 3 = 5.
  • Quadratic Equations: equations in which the highest power of the variable is 2, e.g., x^2 + 4x + 4 = 0.

Methods for Solving Equations

  • Addition/Subtraction: add or subtract the same value to both sides of the equation to isolate the variable.
  • Multiplication/Division: multiply or divide both sides of the equation by the same non-zero value to isolate the variable.
  • Substitution: substitute a value or expression into the equation to solve for the variable.

Inequalities

  • An inequality is a statement that says two mathematical expressions are not equal.
  • It consists of two parts: the left-hand side (LHS) and the right-hand side (RHS), separated by an inequality symbol (, ≤, ≥).
  • The goal is to find the range of values that makes the inequality true.

Types of Inequalities

  • Linear Inequalities: inequalities in which the highest power of the variable is 1, e.g., 2x + 3 > 5.
  • Quadratic Inequalities: inequalities in which the highest power of the variable is 2, e.g., x^2 + 4x + 4 > 0.

Methods for Solving Inequalities

  • Addition/Subtraction: add or subtract the same value to both sides of the inequality to isolate the variable.
  • Multiplication/Division: multiply or divide both sides of the inequality by the same non-zero value, but be careful with the direction of the inequality symbol.
  • Graphing: graph the related function and find the range of values that makes the inequality true.
  • Note: when solving inequalities, the direction of the inequality symbol changes when multiplying or dividing by a negative value.

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