Solving Linear Inequalities

• Mathematics involves the study of topics such as quantity, structure, space, and change.
• Other areas include number theory, algebra, geometry and mathematical analysis.

Linear Inequalities

• A linear inequality is a mathematical statement that compares two expressions using inequality symbols.
• It involves variables raised to the first power.
• Linear inequalities can be represented on a number line or in a coordinate plane.
• Inequality symbols include:
  • < (less than)

  • (greater than)

  • ≤ (less than or equal to)
  • ≥ (greater than or equal to)
• A linear inequality in one variable can be written in the form:
  • ax + b < c
  • ax + b > c
  • ax + b ≤ c
  • ax + b ≥ c
• Where a, b, and c are real numbers and x is the variable.

Solving Linear Inequalities

• Solving linear inequalities involves finding the values of the variable that make the inequality true
• Isolate the variable on one side of the inequality.
• Steps to solve linear inequalities:
  • Simplify both sides of the inequality by distributing and combining like terms.
  • Add or subtract the same number from both sides to isolate the term with the variable.
  • Multiply or divide both sides by the same number to solve for the variable.
  • If you multiply or divide by a negative number, reverse the direction of the inequality sign.
• Example:
  • Solve 3x - 2 > 7
  • Add 2 to both sides: 3x > 9
  • Divide both sides by 3: x > 3
• The solution to a linear inequality is a range of values.
• Graphing the solution on a number line:
  • Use an open circle for < or >.
  • Use a closed circle for ≤ or ≥.
  • Shade the number line in the direction of the solution.
• Compound Inequalities:
  • Two or more inequalities combined into one statement.
  • "And" inequalities: both inequalities must be true.
    • Example: 2 < x < 5
  • "Or" inequalities: at least one inequality must be true.
    • Example: x < 2 or x > 5

Absolute Value Linear Inequalities

• Absolute value represents the distance of a number from zero on the number line.
• Absolute value is always non-negative.
• An absolute value linear inequality involves an absolute value expression compared to a constant using inequality symbols.
• General form:
  • |ax + b| < c
  • |ax + b| > c
  • |ax + b| ≤ c
  • |ax + b| ≥ c
• Where a, b, and c are real numbers and x is the variable.

Solving Absolute Value Linear Inequalities

• Absolute value inequalities can be solved by considering two cases:
  • The expression inside the absolute value is positive or zero.
  • The expression inside the absolute value is negative.
• For |ax + b| < c:
  • Solve -c < ax + b < c.
  • This means ax + b must be between -c and c.
• For |ax + b| > c:
  • Solve ax + b < -c or ax + b > c.
  • This means ax + b must be less than -c or greater than c.
• Steps to solve |ax + b| < c:
  • Set up the compound inequality -c < ax + b < c.
  • Solve for x in the compound inequality.
• Steps to solve |ax + b| > c:
  • Set up the two inequalities ax + b < -c or ax + b > c.
  • Solve each inequality separately.
• Example:
  • Solve |2x - 1| < 5
  • Set up -5 < 2x - 1 < 5
  • Add 1 to all parts: -4 < 2x < 6
  • Divide by 2: -2 < x < 3
• Example:
  • Solve |3x + 2| > 4
  • Set up 3x + 2 < -4 or 3x + 2 > 4
  • Solve 3x + 2 < -4:
    • Subtract 2: 3x < -6
    • Divide by 3: x < -2
  • Solve 3x + 2 > 4:
    • Subtract 2: 3x > 2
    • Divide by 3: x > 2/3
  • Solution: x < -2 or x > 2/3
• When solving absolute value inequalities, it's important to consider the properties of absolute value and to set up the appropriate compound inequalities or separate inequalities.

Absolute Value Linear Equations

• An absolute value linear equation is an equation where the absolute value of a linear expression is equal to a constant.
• General form:
  • |ax + b| = c
• Where a, b, and c are real numbers and x is the variable.
• Absolute value means the distance from zero, so |ax + b| = c implies that ax + b is either c units away from zero in the positive direction or c units away from zero in the negative direction.

Solving Absolute Value Linear Equations

• To solve absolute value equations, consider two cases:
  • The expression inside the absolute value is positive or zero.
  • The expression inside the absolute value is negative.
• For |ax + b| = c, solve two equations:
  • ax + b = c
  • ax + b = -c
• Steps to solve:
  • Set up the two equations without the absolute value:
    • ax + b = c
    • ax + b = -c
  • Solve each equation separately for x.
• Example:
  • Solve |2x - 1| = 5
  • Set up 2x - 1 = 5 and 2x - 1 = -5
  • Solve 2x - 1 = 5:
    • Add 1: 2x = 6
    • Divide by 2: x = 3
  • Solve 2x - 1 = -5:
    • Add 1: 2x = -4
    • Divide by 2: x = -2
  • Solutions: x = 3 and x = -2
• Extraneous Solutions:
  • When solving absolute value equations, it's possible to obtain solutions that do not satisfy the original equation.
  • These are called extraneous solutions.
  • Check all solutions by substituting them back into the original equation to ensure they are valid.
• Example of Extraneous Solutions:
  • Consider |x + 3| = 2x
  • Case 1: x + 3 = 2x, solving gives x = 3. Checking: |3 + 3| = 2(3) -> 6 = 6. Valid.
  • Case 2: x + 3 = -2x, solving gives x = -1. Checking: |-1 + 3| = 2(-1) -> 2 = -2. Invalid.
  • Therefore, x=3 is the only solution. x=-1 is extraneous.
• No Solution:
  • If the absolute value expression is equal to a negative number, there is no solution.
  • Example: |x + 1| = -3 has no solution.