2.3 Solving Equations and Inequalities

• What is the definition of an equation?

  Mathematical statement with equals sign
• Solving an equation means finding the value(s) of the variable that make the equation true.


  True
• Steps for solving equations with variables on both sides

  1️⃣ Move variable terms

  2️⃣ Move constant terms

  3️⃣ Simplify

  4️⃣ Solve

  5️⃣ Check
• The inverse operation of addition is subtraction.


  True
• What is the solution to the equation `5x + 2 = 3x - 4`?

  x = -3
• What operation is used to move variable terms to one side of the equation `5x + 2 = 3x - 4`?

  Subtraction
• When solving equations with parentheses, you must first distribute the number outside the parentheses.


  True
• Solving equations with fractions requires first eliminating the fractions
• What operation is used to eliminate fractions in the equation `1/2 x + 3 = 7`?

  Multiplication
• The least common denominator (LCD) removes the fractions from the equation.
• Multiplying both sides of an equation by the LCD leaves you with an equation with only whole numbers
• To eliminate fractions, multiply both sides by the LCD
• Solving equations with fractions requires an additional step compared to basic equation solving.


  True
• What is the least common denominator in the equation 1/2 x + 3 = 7?

  2
• Steps for solving equations with fractions

  1️⃣ Multiply both sides by the LCD

  2️⃣ Combine like terms

  3️⃣ Isolate the variable

  4️⃣ Check the solution
• The key to solving equations with fractions is to eliminate the fractions by multiplying by the LCD
• Inverse operations are used to isolate variables in both equations and inequalities.


  True
• The solution to an inequality is a range of values rather than a single value.

  True
• Solve the equation x - 5 = 2

  x = 7
• Moving smaller variable terms to larger ones helps avoid negative coefficients.


  True
• Steps for solving equations with parentheses

  1️⃣ Expand parentheses

  2️⃣ Simplify by combining like terms

  3️⃣ Use inverse operations to solve

  4️⃣ Check the solution
• To expand parentheses, multiply the term outside by each term inside
• What is the key first step in solving equations with fractions?

  Eliminate the fractions
• What is the difference between solving basic equations and equations with fractions?

  Fractions need elimination
• What is the first step when solving equations with variables on both sides?

  Move variable terms
• What is the solution to the equation `2x = -6`?

  x = -3
• The key difference from basic equation solving is the need to first rearrange the equation to group like terms
• Steps for solving equations with variables on both sides

  1️⃣ Move variable terms to one side

  2️⃣ Move constant terms to the opposite side

  3️⃣ Simplify by combining like terms

  4️⃣ Solve by using inverse operations

  5️⃣ Check the solution
• Solving equations with parentheses involves removing them by distributing the number outside to the terms inside
• What is the result of expanding the parentheses in the equation `3(2x + 1) = 15`?

  6x + 3 = 15
• Steps for solving equations with fractions

  1️⃣ Multiply both sides by the LCD

  2️⃣ Combine like terms on each side

  3️⃣ Solve by inverse operations

  4️⃣ Check the solution
• Why is it necessary to eliminate fractions before solving an equation with fractions?

  To simplify the equation
• What is the LCD in the equation `1/2 x + 3 = 7`?

  2
• What is the solution to the equation 1/2 x + 3 = 7?

  x = 8
• Match the inequality symbol with its meaning:

  >> ↔️ Greater than

  < ↔️ Less than

  ≥> ↔️ Greater than or equal to

  ≤> ↔️ Less than or equal to
• To solve a simple inequality, the first step is to isolate the variable term.
• The inverse operations used to isolate the variable are addition/subtraction and multiplication/division, just like when solving equations
• An equation is a mathematical statement with an equals sign
• Solve the equation 3x + 4 = 19

  x = 5
• Solve the equation 5x + 2 = 3x - 4

  x = -3