Solving Inequalities and Graphing on a Number Line

Key Concepts

• Solving inequalities involves similar steps as solving equations.
• The main difference is plotting the solution on a number line.
• While solving:
  • Treat the inequality sign like an equal sign.
  • When multiplying or dividing by a negative number, reverse the inequality sign.

Example 1: (2x + 3 > 7)

1. Subtract 3 from both sides:
   • (2x > 4)
2. Divide both sides by 2:
   • (x > 2)
3. Graph:
   • Open circle at 2, shade to the right.
4. Interval Notation:
   • ((2, \infty))

Example 2: (\frac{1}{3}x + 4 \leq 6)

1. Subtract 4 from both sides:
   • (\frac{1}{3}x \leq 2)
2. Multiply both sides by 3 to clear the fraction:
   • (x \leq 6)
3. Graph:
   • Closed circle at 6, shade to the left.
4. Interval Notation:
   • ((-\infty, 6])

Example 3: (-3x - 5 > -9)

1. Add 5 to both sides:
   • (-3x > -4)
2. Divide by -3, reverse inequality sign:
   • (x < 3)
3. Graph:
   • Closed circle at 3, shade to the left.
4. Interval Notation:
   • ((-\infty, 3])

Example 4: (2x - 1 > 7) or (-3x + 2 \geq -1)

Solving

• First Equation:
  1. Add 1 to both sides: (2x > 8)
  2. Divide by 2: (x > 4)
  3. Open circle at 4, shade to the right.
• Second Equation:
  1. Subtract 2: (-3x \geq -3)
  2. Divide by -3, reverse inequality: (x \leq 1)
  3. Closed circle at 1, shade to the left.

Interval Notation

• ((-\infty, 1] \cup (4, \infty))

Example 5: Compound Inequality (-12 < 7x - 5 \leq 9)

1. Add 5 to all parts:
   • (-7 < 7x \leq 14)
2. Divide all parts by 7:
   • (-1 < x \leq 2)
3. Graph:
   • Open circle at -1, closed circle at 2, shade in between.
4. Interval Notation:
   • ((-1, 2])

Important Notes

• Open circles indicate the number is not included in the solution ((<) or (>)).
• Closed circles indicate the number is included ((\leq) or (\geq)).
• Always use parentheses with infinity symbols in interval notation.