Inequalities and Absolute Value Equations and
Inequations - §2.4 - 2.5
Fall 2013 - Math 1010
A no TIE fighter and squinting cat zone.
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Roadmap
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Notes for solving inequalities.
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Examples of solving inequalities.
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Notes for solving absolute value equations and inequalites.
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Examples of solving absolute value equations and inequalities.
Today’s lecture will procede with slide notes and then chalkboard
examples.
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§2.4 - The Symbols for Inequalities
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’less than’ Example x < 7
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’less than or equal’ Example x ≤ 4
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’greater than’ Example x > −3
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’greater than or equal’ Example x ≥ 2.8
- ’or equal’ means the values for the unknown can include that number.
- Inequalties without ’or equal’, that is < and >, are called strict
inequalities.
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§2.4 - Notations
Styles in this section include intervals, inequalty signs, graphs, and set (or
set builder) notation. The samples below are equivalent.
Example x + 6 < 9
↔x <3
Interval: (−∞, 3)
Inequality signs: x < 3
Set builder notation: {x|x < 3} (I like to use : in place of |.)
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§2.4 - Operations
Operations on all sides of an inequality with expressions are the same for
equalities with one exception:
Multiplication and division by a negative quantity produces an
equivalent inequality with a reversed inequality symbol.
Summary:
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Add/Subtract: a < b ↔ a + c < b + c ↔ a − c < b − c
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Multiply/Divide: positive quantities
a < b ↔ ac < bc ↔ a ÷ c < b ÷ c, c > 0
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Multiply/Divide: negative quantities
a < b ↔ ac > bc ↔ a ÷ c > b ÷ c, c < 0
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Transitivity: When a < b and b < c, then it follows that a < c.
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§2.4 - Compound Inequalties - Conjunctive ’And’
Conjunctive (and) is used for compound inequalties that have two
conditions. Both conditions must be met.
Example −1 ≤ 5 − 2x and 5 − 2x < 7 Write this as a double inequalty
and solve.
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§2.4 - Compound Inequalties - Conjunctive ’And’
Conjunctive (and) is used for compound inequalties that have two
conditions. Both conditions must be met.
Example −1 ≤ 5 − 2x and 5 − 2x < 7 Write this as a double inequalty
and solve.
−1 ≤ 5 − 2x < 7
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§2.4 - Compound Inequalties - Conjunctive ’And’
Conjunctive (and) is used for compound inequalties that have two
conditions. Both conditions must be met.
Example −1 ≤ 5 − 2x and 5 − 2x < 7 Write this as a double inequalty
and solve.
−1 ≤ 5 − 2x < 7
−6 ≤ −2x < 2
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§2.4 - Compound Inequalties - Conjunctive ’And’
Conjunctive (and) is used for compound inequalties that have two
conditions. Both conditions must be met.
Example −1 ≤ 5 − 2x and 5 − 2x < 7 Write this as a double inequalty
and solve.
−1 ≤ 5 − 2x < 7
−6 ≤ −2x < 2
−6
2
≥x >
−2
−2
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§2.4 - Compound Inequalties - Conjunctive ’And’
Conjunctive (and) is used for compound inequalties that have two
conditions. Both conditions must be met.
Example −1 ≤ 5 − 2x and 5 − 2x < 7 Write this as a double inequalty
and solve.
−1 ≤ 5 − 2x < 7
−6 ≤ −2x < 2
−6
2
≥x >
−2
−2
3 ≥ x > −1 ↔ −1 < x ≤ 3
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Just Say No
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§2.4 - Compound Inequalties - Disjunctive ’Or’
Disjuctive (or) is used for compound inequalties that have two conditions.
Either condition may be met. These inequalities cannot be written as a
compound inequality.
Example x + 3 < −7 or x + 3 > 14 Solve.
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§2.4 - Compound Inequalties - Disjunctive ’Or’
Disjuctive (or) is used for compound inequalties that have two conditions.
Either condition may be met. These inequalities cannot be written as a
compound inequality.
Example x + 3 < −7 or x + 3 > 14 Solve.
One at a time: x + 3 < −7 → x < −10
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§2.4 - Compound Inequalties - Disjunctive ’Or’
Disjuctive (or) is used for compound inequalties that have two conditions.
Either condition may be met. These inequalities cannot be written as a
compound inequality.
Example x + 3 < −7 or x + 3 > 14 Solve.
One at a time: x + 3 < −7 → x < −10
Next one: x + 3 > 14 → x > 11
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§2.4 - Compound Inequalties - Disjunctive ’Or’
Disjuctive (or) is used for compound inequalties that have two conditions.
Either condition may be met. These inequalities cannot be written as a
compound inequality.
Example x + 3 < −7 or x + 3 > 14 Solve.
One at a time: x + 3 < −7 → x < −10
Next one: x + 3 > 14 → x > 11
Solution: x < −10 or x > 11.
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§2.5 - Absolute Value Equations and Inequalties
An absolute value equation contains a term with an absolute value
expression. It may have no solution, or it may have one or more solutions.
Example |x| = 3.6 → x = −3.6 or x = 3.6
Example
|x| = −12 → No solution; an absolute value cannot be negative.
To solve |x| = a, a ≥ 0 the algebraic expression inside the absolute value
symbols x may be −a or a.
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§2.5 - Solving Equations
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Simplify each side to have at most one absolute value expression.
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Write two linear equations - write one side equal to the other side,
and then write one side equal to the opposite of the other side.
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Solve each linear equation one after the other.
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Check the solutions! It is possible for no solution or one solution.
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§2.5 - Solving Inequalities
Absolute value inequalties match a conjunctive pair of statements (also, a
compound inequality) or a disjunctive pair of statements.
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§2.5 - Solving Inequalities
Absolute value inequalties match a conjunctive pair of statements (also, a
compound inequality) or a disjunctive pair of statements.
Example Solutions to |x| < 2 lie between -2 and 2. That is, −2 < x < 2.
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§2.5 - Solving Inequalities
Absolute value inequalties match a conjunctive pair of statements (also, a
compound inequality) or a disjunctive pair of statements.
Example Solutions to |x| < 2 lie between -2 and 2. That is, −2 < x < 2.
Example Solutions to |x| > 2 lie outside -2 and 2. That is, x < −2 or
x > 2.
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Assignment
Assignment:
For Monday:
1. Read sections 3.1 and 3.2.
2. Exercises from §2.4, 2.5 due Monday, September 16.
3. Pre-Exam 1 Wednesday, September 18.
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