Math  201819
8.18  Inequalities
8.18 The student will solve multistep linear inequalities in one variable with the variable on one or both sides of the inequality symbol, including practical problems, and graph the solution on a number line.
BIG IDEAS
 I can find how far and how fast a bus travels, the number
and combinations of fruits that can be purchased, and how long it takes to
drain a swimming pool. I can avoid
receiving a speeding ticket when I can drive and won’t go over the allowed
number of text messages per month on my cell phone.
 I will
be able to write symbolic representations of the way
numbers behave and will know that in order to maintain equality, an
operation performed on one side must also be performed on the other side.
UNDERSTANDING THE STANDARD
·
A multistep
inequality may include, but not be limited to inequalities such as the
following:
> ; ;
· When both expressions of an inequality are multiplied or divided by a negative number, the inequality sign reverses.
· A solution to an inequality is the value or set of values that can be substituted to make the inequality true.
· In an inequality, there can be more than one value for the variable that makes the inequality true. There can be many solutions. (i.e., x + 4 > −3 then the solutions is x > −7. This means that x can be any number greater than −7. A few solutions might be −6.5, −3, 0, 4, 25, etc.)
· Realworld problems can be modeled and solved using linear inequalities.
· The properties of real numbers and properties of inequality can be used to solve inequalities, justify solutions, and express simplification. Students should use the following properties, where appropriate, to further develop flexibility and fluency in problem solving (limitations may exist for the values of a, b, or c in this standard).
 Commutative property of addition: .
 Commutative property of multiplication: .
 Associative property of addition: .
 Associative property of multiplication: .
 Subtraction and division are neither commutative nor associative.

Distributive
property (over addition/subtraction):
.
 The additive identity is zero (0) because any number added to zero is the number. The multiplicative identity is one (1) because any number multiplied by one is the number. There are no identity elements for subtraction and division.
 Identity property of addition (additive identity property): .
 Identity property of multiplication (multiplicative identity property): .

Inverses are
numbers that combine with other numbers and result in identity elements
(e.g., 5 + (–5) = 0; · 5 = 1).
 Inverse property of addition (additive inverse property): .
 Inverse property of multiplication (multiplicative inverse property): .
 Zero has no multiplicative inverse.
 Multiplicative property of zero: .
 Division by zero is not a possible mathematical operation. It is undefined.
 Substitution property: If , then b can be substituted for a in any expression, equation, or inequality.
 Addition property of inequality: If then; if then .
 Subtraction property of inequality: If then; if then .

Multiplication
property of inequality: If then ; if
then .

Multiplication
property of inequality (multiplication by a negative number): If
then ; if then .
 Division property of inequality: If then ; if then .
 Division property of inequality (division by a negative number): If then ; if then .ESSENTIALS
 How does the
solution to an equation differ from the solution to an inequality?
While a linear equation has only one replacement value for the variable that makes the equation true, an inequality can have more than one.
The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to
· 8.18_{3} Write verbal expressions and sentences as algebraic expressions and inequalities.
· 8.18_{4} Write algebraic expressions and inequalities as verbal expressions and sentences.
·
8.18_{1} Apply
properties of real numbers and properties of inequality to solve multistep linear inequalities (up to four steps) in one
variable with the variable on one or both sides of the inequality. Coefficients
and numeric terms will be rational.
Inequalities may contain expressions that need to be expanded (using the
distributive property) or require collecting like terms to solve.
· 8.18_{5} Solve practical problems that require the solution of a multistep linear inequality in one variable.
·
8.18_{2} Graph
solutions to multistep linear inequalities on a number line.
· 8.18_{6} Identify a numerical value(s) that is part of the solution set of a given inequality.
KEY VOCABULARY
linear equation, pictorial representations, two step inequalities, algebraic sentences, commutative property of addition and multiplication, associative property of addition and multiplication, distributive property, identity property of addition and multiplication, zero property of multiplication, additive inverse, multiplicative inverse