# 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

2016 VDOE Curriculum Framework - 8.18 Understanding

·  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.)

·  Real-world 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.

.

-  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 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 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

·  8.183  Write verbal expressions and sentences as algebraic expressions and inequalities.

·  8.184  Write algebraic expressions and inequalities as verbal expressions and sentences.

·  8.181  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.

·  8.185  Solve practical problems that require the solution of a multistep linear inequality in one variable.

·  8.182  Graph solutions to multistep linear inequalities on a number line.

·  8.186  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

Updated: Jul 30, 2019