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Graphing and Solving Inequalities Guide
Sep 7, 2024
Graphing and Solving Inequalities
Key Concepts
Representing inequalities on number lines
Shading directions based on inequality signs
Intervals and interval notation
Solving inequalities with fractions, variables, and absolute values
Graphing Inequalities on Number Lines
Example 1: x > 2
Open circle
at 2 (no underline)
Shade to the
right
(greater than)
Interval Notation
: (2, ∞)
Example 2: x ≥ -1
Closed circle
at -1 (includes -1)
Shade to the
right
(greater than or equal to)
Interval Notation
: [-1, ∞)
Example 3: x < 4
Open circle
at 4
Shade to the
left
(less than)
Interval Notation
: (-∞, 4)
Example 4: x ≤ -2
Closed circle
at -2
Shade to the
left
(less than or equal to)
Interval Notation
: (-∞, -2]
Example 5: 1 < x ≤ 4
Open circle
at 1 and
closed circle
at 4
Shade
between
1 and 4
Interval Notation
: (1, 4]
Example 6: x < -2 or x ≥ 3
Open circle
at -2 and
closed circle
at 3
Shade
left
of -2 and
right
of 3
Interval Notation
: (-∞, -2) ∪ [3, ∞)
Solving Inequalities
Example 1: x + 4 > 5
Subtract 4: x > 1
Interval Notation
: (1, ∞)
Example 2: -2x + 5 < 12
Subtract 5: -2x < 7
Divide by -2 (flip inequality): x > -3.5
Interval Notation
: (-3.5, ∞)
Example 3: 7 - 2x ≤ 12
Subtract 7: -2x ≤ 5
Divide by -2 (flip inequality): x ≥ -2.5
Interval Notation
: [-2.5, ∞)
Example 4: 4 - 2x ≥ 3x + 19
Move terms: -5x ≥ 15
Divide by -5 (flip inequality): x ≤ -3
Interval Notation
: (-∞, -3]
Example 5: 1/3x + 4 > 8
Subtract 4: 1/3x > 4
Multiply by 3: x > 12
Interval Notation
: (12, ∞)
Solving Inequalities with Absolute Value
Example 1: |x| < 4
Two equations: x < 4 and x > -4
Interval Notation: (-4, 4)
Example 2: |2x + 3| ≥ 8
Two equations: 2x + 3 ≥ 8 and 2x + 3 ≤ -8
Solve both:
2x ≥ 5 → x ≥ 2.5
2x ≤ -11 → x ≤ -5.5
Interval Notation
: (-∞, -5.5] ∪ [2.5, ∞)
Example 3: 5 - 3|4x + 1| ≥ -9
Move terms: |4x + 1| ≤ 14/3
Write two equations: 4x + 1 ≤ 14/3 and 4x + 1 ≥ -14/3
Solve both:
4x ≤ 11/3 → x ≤ 11/12
4x ≥ -17/3 → x ≥ -17/12
Interval Notation
: [-17/12, 11/12]
Summary
Understand different types of inequalities and how to graph and solve them.
Be familiar with using interval notation to express results succinctly.
Practice solving inequalities in different forms including those with fractions and absolute values.
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