Patterns of Special Products · Inequalities

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Number and Quantity/Algebra · 6 min read

Patterns of Special Products · Inequalities

Special products are algebraic patterns that allow efficient multiplication without distributing term by term.

Square of a Binomial

(a + b)² = a² + 2ab + b²

This pattern comes from (a + b)(a + b) using FOIL: First: a² Outer: ab Inner: ab Last: b² Sum: a² + 2ab + b²

Worked Example

(x + 5)² = x² + 2(x)(5) + 25 = x² + 10x + 25

(a − b)² = a² − 2ab + b²

Worked Example

(3x − 4)² = 9x² − 24x + 16

Key Point

Common Error: (a + b)² ≠ a² + b². The middle term 2ab must be included.

Difference of Squares

(a + b)(a − b) = a² − b²

The outer and inner terms cancel because ab − ab = 0.

Worked Example

(x + 7)(x − 7) = x² − 49

This pattern is useful for factoring: x² − 16 = (x + 4)(x − 4) 4y² − 25 = (2y + 5)(2y − 5)

Note: Sum of squares a² + b² does not factor over the real numbers.

Cube of a Binomial

(a + b)³ = a³ + 3a²b + 3ab² + b³

(a − b)³ = a³ − 3a²b + 3ab² − b³

Worked Example

(x + 2)³ = x³ + 3(x²)(2) + 3(x)(4) + 8 = x³ + 6x² + 12x + 8

Sum and Difference of Cubes

Sum of Cubes

a³ + b³ = (a + b)(a² − ab + b²)

Difference of Cubes

a³ − b³ = (a − b)(a² + ab + b²)

Mnemonic

"SOAP" – Same sign, Opposite sign, Always Positive

Worked Example

x³ + 27 = x³ + 3³ = (x + 3)(x² − 3x + 9) Example: 8y³ − 125 = (2y)³ − 5³ = (2y − 5)(4y² + 10y + 25)

Product of Binomials (General)

(ax + b)(cx + d) = acx² + (ad + bc)x + bd

Worked Example

(2x + 3)(4x − 5) = (2)(4)x² + [(2)(−5) + (3)(4)]x + (3)(−5) = 8x² + (−10 + 12)x − 15 = 8x² + 2x − 15

Recognizing Patterns for Factoring

When factoring expressions, recognize these patterns in reverse:

Expression Form	Factored Form
a² + 2ab + b²	(a + b)²
a² − 2ab + b²	(a − b)²
a² − b²	(a + b)(a − b)
a³ + b³	(a + b)(a² − ab + b²)
a³ − b³	(a − b)(a² + ab + b²)

Inequalities compare expressions using symbols indicating relative size rather than equality.

Inequality Symbols

a < b means "a is less than b"

a > b means "a is greater than b"

a ≤ b means "a is less than or equal to b"

a ≥ b means "a is greater than or equal to b"

a ≠ b means "a is not equal to b"

Properties of Inequalities

Transitive Property

If a < b and b < c, then a < c.

Addition/Subtraction Property

Adding or subtracting the same quantity from both sides preserves the inequality direction. If a < b, then a + c < b + c and a − c < b − c.

Multiplication/Division Property (Positive Multiplier)

Multiplying or dividing both sides by a positive number preserves the direction. If a < b and c > 0, then ac < bc.

Multiplication/Division Property (Negative Multiplier)

Multiplying or dividing both sides by a negative number reverses the inequality direction. If a < b and c < 0, then ac > bc.

This reversal is critical and a common source of errors.

Solving Linear Inequalities

Solve like linear equations, but reverse the inequality when multiplying or dividing by a negative number.

Worked Example

Solve 3x − 7 ≤ 14 3x ≤ 21 x ≤ 7 Solution: (−∞, 7] or x ≤ 7
Solve −4x + 5 > 17 −4x > 12 x < −3 (inequality reversed due to division by −4) Solution: (−∞, −3) or x < −3

Compound Inequalities

Conjunction (AND): Both conditions must be satisfied simultaneously. Example: −3 < 2x + 1 ≤ 9 Solve as two inequalities: −3 < 2x + 1 → −4 < 2x → −2 < x 2x + 1 ≤ 9 → 2x ≤ 8 → x ≤ 4 Combined: −2 < x ≤ 4 or (−2, 4]

Disjunction (OR): At least one condition must be satisfied. Example: x − 2 < −5 OR x − 2 > 5 x < −3 OR x > 7 Solution: (−∞, −3) ∪ (7, ∞)

Absolute Value Inequalities

For a > 0:

|x| < a means −a < x < a (AND condition)

|x| > a means x < −a OR x > a (OR condition)

Worked Example

|2x − 3| < 7 −7 < 2x − 3 < 7 −4 < 2x < 10 −2 < x < 5 Solution: (−2, 5)
|x + 4| ≥ 2 x + 4 ≤ −2 OR x + 4 ≥ 2 x ≤ −6 OR x ≥ −2 Solution: (−∞, −6] ∪ [−2, ∞)

Quadratic Inequalities

To solve ax² + bx + c > 0 (or <, ≤, ≥):

Find zeros by solving ax² + bx + c = 0

Plot zeros on a number line

Test points in each interval

Identify intervals satisfying the inequality

Worked Example

x² − 5x + 6 > 0 Factor: (x − 2)(x − 3) > 0 Zeros: x = 2 and x = 3 Test intervals:
x < 2: Choose x = 0 → (−2)(−3) = 6 > 0
2 < x < 3: Choose x = 2.5 → (0.5)(−0.5) = −0.25 < 0
x > 3: Choose x = 4 → (2)(1) = 2 > 0

Solution: (−∞, 2) ∪ (3, ∞)

Rational Inequalities

For inequalities involving fractions:

Move all terms to one side

Find common denominator

Determine critical values (zeros of numerator and denominator)

Create sign chart

Select intervals satisfying the inequality

Important: Values making the denominator zero are excluded from the solution.

Worked Example

(x − 1)/(x + 2) ≥ 0

Critical values: x = 1 (numerator zero), x = −2 (denominator zero, excluded)

Test intervals:

x < −2

Both negative → positive quotient

−2 < x < 1

Numerator negative, denominator positive → negative

x > 1

Both positive → positive

At x = 1

Expression equals 0, which satisfies ≥ 0

Solution

(−∞, −2) ∪ [1, ∞)

Graphing Inequalities

One Variable: Use a number line with:

Open circle for < or > (boundary not included)

Closed circle for ≤ or ≥ (boundary included)

Shade the solution region

Two Variables: Graph the boundary line, then shade the half-plane containing solutions.

Dashed line for < or >

Solid line for ≤ or ≥

Test a point to determine which side to shade

Finished reading?

Lock it in with a quick Number and Quantity/Algebra quiz.

Quiz me Next chapter →

← Multiplying and Dividing Signed Numbers · Properties of Natural Numbers

Patterns of Special Products · Inequalities

Special products are algebraic patterns that allow efficient multiplication without distributing term by term.

Square of a Binomial

(a + b)² = a² + 2ab + b²

This pattern comes from (a + b)(a + b) using FOIL: First: a² Outer: ab Inner: ab Last: b² Sum: a² + 2ab + b²

Worked Example

(x + 5)² = x² + 2(x)(5) + 25 = x² + 10x + 25

(a − b)² = a² − 2ab + b²

Worked Example

(3x − 4)² = 9x² − 24x + 16

Key Point

Common Error: (a + b)² ≠ a² + b². The middle term 2ab must be included.

Difference of Squares

(a + b)(a − b) = a² − b²

The outer and inner terms cancel because ab − ab = 0.

Worked Example

(x + 7)(x − 7) = x² − 49

This pattern is useful for factoring: x² − 16 = (x + 4)(x − 4) 4y² − 25 = (2y + 5)(2y − 5)

Note: Sum of squares a² + b² does not factor over the real numbers.

Cube of a Binomial

(a + b)³ = a³ + 3a²b + 3ab² + b³

(a − b)³ = a³ − 3a²b + 3ab² − b³

Worked Example

(x + 2)³ = x³ + 3(x²)(2) + 3(x)(4) + 8 = x³ + 6x² + 12x + 8

Sum and Difference of Cubes

Sum of Cubes

a³ + b³ = (a + b)(a² − ab + b²)

Difference of Cubes

a³ − b³ = (a − b)(a² + ab + b²)

Mnemonic

"SOAP" – Same sign, Opposite sign, Always Positive

Worked Example

x³ + 27 = x³ + 3³ = (x + 3)(x² − 3x + 9) Example: 8y³ − 125 = (2y)³ − 5³ = (2y − 5)(4y² + 10y + 25)

Product of Binomials (General)

(ax + b)(cx + d) = acx² + (ad + bc)x + bd

Worked Example

(2x + 3)(4x − 5) = (2)(4)x² + [(2)(−5) + (3)(4)]x + (3)(−5) = 8x² + (−10 + 12)x − 15 = 8x² + 2x − 15

Recognizing Patterns for Factoring

When factoring expressions, recognize these patterns in reverse:

Expression Form	Factored Form
a² + 2ab + b²	(a + b)²
a² − 2ab + b²	(a − b)²
a² − b²	(a + b)(a − b)
a³ + b³	(a + b)(a² − ab + b²)
a³ − b³	(a − b)(a² + ab + b²)

Inequalities compare expressions using symbols indicating relative size rather than equality.

Inequality Symbols

a < b means "a is less than b"

a > b means "a is greater than b"

a ≤ b means "a is less than or equal to b"

a ≥ b means "a is greater than or equal to b"

a ≠ b means "a is not equal to b"

Properties of Inequalities

Transitive Property

If a < b and b < c, then a < c.

Addition/Subtraction Property

Adding or subtracting the same quantity from both sides preserves the inequality direction. If a < b, then a + c < b + c and a − c < b − c.

Multiplication/Division Property (Positive Multiplier)

Multiplying or dividing both sides by a positive number preserves the direction. If a < b and c > 0, then ac < bc.

Multiplication/Division Property (Negative Multiplier)

Multiplying or dividing both sides by a negative number reverses the inequality direction. If a < b and c < 0, then ac > bc.

This reversal is critical and a common source of errors.

Solving Linear Inequalities

Solve like linear equations, but reverse the inequality when multiplying or dividing by a negative number.

Worked Example

Solve 3x − 7 ≤ 14 3x ≤ 21 x ≤ 7 Solution: (−∞, 7] or x ≤ 7
Solve −4x + 5 > 17 −4x > 12 x < −3 (inequality reversed due to division by −4) Solution: (−∞, −3) or x < −3

Compound Inequalities

Disjunction (OR): At least one condition must be satisfied. Example: x − 2 < −5 OR x − 2 > 5 x < −3 OR x > 7 Solution: (−∞, −3) ∪ (7, ∞)

Absolute Value Inequalities

For a > 0:

|x| < a means −a < x < a (AND condition)

|x| > a means x < −a OR x > a (OR condition)

Worked Example

|2x − 3| < 7 −7 < 2x − 3 < 7 −4 < 2x < 10 −2 < x < 5 Solution: (−2, 5)
|x + 4| ≥ 2 x + 4 ≤ −2 OR x + 4 ≥ 2 x ≤ −6 OR x ≥ −2 Solution: (−∞, −6] ∪ [−2, ∞)

Quadratic Inequalities

To solve ax² + bx + c > 0 (or <, ≤, ≥):

Find zeros by solving ax² + bx + c = 0

Plot zeros on a number line

Test points in each interval

Identify intervals satisfying the inequality

Worked Example

x² − 5x + 6 > 0 Factor: (x − 2)(x − 3) > 0 Zeros: x = 2 and x = 3 Test intervals:
x < 2: Choose x = 0 → (−2)(−3) = 6 > 0
2 < x < 3: Choose x = 2.5 → (0.5)(−0.5) = −0.25 < 0
x > 3: Choose x = 4 → (2)(1) = 2 > 0

Solution: (−∞, 2) ∪ (3, ∞)

Rational Inequalities

For inequalities involving fractions:

Move all terms to one side

Find common denominator

Determine critical values (zeros of numerator and denominator)

Create sign chart

Select intervals satisfying the inequality

Important: Values making the denominator zero are excluded from the solution.

Worked Example

(x − 1)/(x + 2) ≥ 0

Critical values: x = 1 (numerator zero), x = −2 (denominator zero, excluded)

Test intervals:

x < −2

Both negative → positive quotient

−2 < x < 1

Numerator negative, denominator positive → negative

x > 1

Both positive → positive

At x = 1

Expression equals 0, which satisfies ≥ 0

Solution

(−∞, −2) ∪ [1, ∞)

Graphing Inequalities

One Variable: Use a number line with:

Open circle for < or > (boundary not included)

Closed circle for ≤ or ≥ (boundary included)

Shade the solution region

Two Variables: Graph the boundary line, then shade the half-plane containing solutions.

Dashed line for < or >

Solid line for ≤ or ≥

Test a point to determine which side to shade

Finished reading?

Lock it in with a quick Number and Quantity/Algebra quiz.

Quiz me Next chapter →

NYSTCE Mathematics CST (004)