Expressions and Like Terms

Creating and Evaluating Expressions

• An expression involves numbers, variables, and operations as a mathematical phrase.
• Evaluating expressions involves substituting values for variables and simplifying the expression.
• Steps to evaluate expressions:
  • Identify given variable values.
  • Substitute values into the expression.
  • Perform operations following the correct order (BEDMAS: Brackets, Exponents, Division/Multiplication, Addition/Subtraction).
• Example:
  • If x = 3, evaluate 2x + 5.
  • Substitute x = 3 to get 2(3) + 5.
  • Multiply to get 6 + 5.
  • The final answer is 11.

Collecting Like Terms

• Like terms refers to terms sharing the same variable raised to the same exponent.
• Combine like terms' coefficients, keeping variables unchanged.
• Steps to collect like terms:
  • Identify terms sharing the same variable.
  • Add or subtract coefficients without changing the variable.
  • Simplify the expression.
• Example: 3x + 5x - 2 + 7
  • Like terms: 3x and 5x, -2 and 7
  • Add like terms: (3x + 5x) + (-2 + 7) = 8x + 5

Solving by Inspection

• Solving simple equations by logically thinking about the answer
• Steps to solve by inspection:
  • Determine the number that makes the equation true.
  • Check the answer by substituting into the equation.
• Example: Solve x + 4 = 10
  • What number plus 4 equals 10? Answer: x = 6
  • Check: 6 + 4 = 10 (Correct!).

Solving by Systematic Trial

• Testing different values until the correct solution is found
• Steps to solve by systematic trial:
  • Choose a value for the variable.
  • Substitute the value into the equation & check if both sides are equal.
  • Adjust the value and repeat until the equation is true.
• Example: Solve 2x + 3 = 9
  • Try x = 2 → 2(2) + 3 = 4 + 3 = 7 (too low).
  • Try x = 3 → 2(3) + 3 = 6 + 3 = 9 (Correct!).

Solving by Isolation

• Isolating the variable needs performing inverse operations.
• Steps to solve by isolation:
  • Identify the operations applied to the variable.
  • Undo these operations by using inverse operations, step by step.
  • Keep both sides of the equation balanced.
• Example: Solve 3x - 5 = 10
  • Add 5 to both sides → 3x = 15
  • Divide by 3 → x = 5.

Solving Multi-Step Equations

• Follow multiple steps to isolate the variable.
• Steps to solve multi-step equations:
  • Simplify both sides of the equation if needed (combine like terms).
  • Use inverse operations to move terms to one side.
  • Isolate the variable.
• Example: Solve 2x + 3 = 4x - 5
  • Subtract 2x from both sides → 3 = 2x - 5.
  • Add 5 to both sides → 8 = 2x.
  • Divide by 2 → x = 4.

Graphing Inequalities

• A number line can show the solution set to inequalities
• Steps to graph inequalities:
  • Identify the inequality sign (<, >, ≤, ≥)
  • Use an open circle (○) for < or >, and a closed circle (•) for ≤ or ≥.
  • Shade the side where values satisfy the inequality.
• Example: Graph x > 2
  • Open circle at 2, shade all numbers greater than 2.

Solving Inequalities

• Inequalities are solved like equations, but remember to flip the inequality sign when multiplying or dividing by a negative number.
• Steps to solve inequalities:
  • Solve as you would an equation.
  • Reverse the inequality sign if multiplying/dividing by a negative.
• Example: Solve -2x > 6
  • Divide by -2 and flip the sign!
  • x < -3
  • Graph: Open circle at -3, shade left.