Linear Functions, Equations, and Inequalities

UNIT OVERVIEW

GOALS AND STANDARDS

MATHEMATICS BACKGROUND

UNIT INTRODUCTION

Mathematics Background

Linear Functions, Equations, and Inequalities

Although the basic understandings of and skills for linear equations were addressed in the Grade 7 Unit Moving Straight Ahead, they need to be revisited and practiced to deepen student understanding. The Problems in Investigation 2 of Thinking With Mathematical Models are designed to promote this sort of review and extension.

Linear Functions and Equations

In Moving Straight Ahead, students learned to recognize, represent symbolically, and analyze relationships in which a dependent variable changes at a constant rate relative to an independent variable.

Students learned the connections between the equation y = mx + b, the rate of change, the slope of the line, and the y-intercept of the line.

y = mx + b

y-intercept, or constant term, b

y-intercept

y

(0, b)

Slope

=

m

=

change in y change in x

change in y (rise)

change in x (run)

O

x

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Solving Equations

Many questions about linear functions can be answered by solving equations of the form c = mx + b for x (in which c is a constant).

In Moving Straight Ahead, students learned to approximate solutions to such

equations by using tables and graphs of (x, y) values. They also learned to find

exact

solutions

by

reversing

the

operations

to

get

x

=

(c

- b) m

and

by

using

the

properties of equality.

Properties of Equality

Addition Property of Equality

If you add the same number to each side of an equation, the two sides remain equal.

Arithmetic

Algebra

10 = 5(2), so 10 + 3 = 5(2) + 3.

If a = b, then a + c = b + c.

Subtraction Property of Equality

If you subtract the same number from each side of an equation, the two sides remain equal.

Arithmetic

Algebra

10 = 5(2), so 10 - 3 = 5(2) - 3.

If a = b, then a - c = b - c.

Multiplication Property of Equality

If you multiply each side of an equation by the same number, the two sides remain equal.

Arithmetic

Algebra

10 = 5(2), so 10 ? 3 = 5(2) ? 3.

If a = b, then a ? c = b ? c.

Division Property of Equality

If you divide each side of an equation by the same number, the two sides remain equal.

Arithmetic

Algebra

10 = 5(2), so 10 ? 3 = 5(2) ? 3.

If a = b, then a ? c = b ? c.

Fact Families

Another way to solve equations is to use fact families.

Fact families with whole-number operations are introduced in Grade 6, and students revisit them in the Grade 7 Unit Accentuate the Negative. The concept of fact families highlights the relationships between addition and subtraction and between multiplication and division. Students can interpret subtraction problems as missing addend problems and division problems as missing factor problems.

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UNIT OVERVIEW

GOALS AND STANDARDS

MATHEMATICS BACKGROUND

UNIT INTRODUCTION

To solve c = mx + b for x, look at its fact family.

There are three equations in the addition and subtraction fact family for c = mx + b.

c = mx + b

c ? b = mx

c ? mx = b

To solve c = mx + b for x, choose the equivalent equation c - b = mx.

There are three equations in the multiplication and division fact family for c ? b = mx.

c ? b = mx

(c

? x

b)

=

m

(c

? b) m

=

x

To

solve

c

-

b

=

mx

for

x,

choose

the

equivalent

equation

(c

m

b)

=

x.

So

the

solution

of

c

=

mx

+

b

is

x

=

(c

m

b)

.

Inequalities

Many real-world problems involve inequalities rather than equations. Inequalities are mathematical sentences that use ... , ? , 6 , or 7 , such as c ... mx + b or c ? mx + b.

Problems in Thinking With Mathematical Models invite students to recognize the implications of phrases such as "at least" and "at most," to use inequality notation for problem conditions, and to use tables and graphs for finding solutions of inequalities. Some inequalities have infinitely many solutions. Some inequalities, such as x2 ... 0, have one solution. Other inequalities, such as x2 6 0, have no solution. The table shows that 1.5x + 1 7 7 for x 7 4.

x

y

-1

- 0.5

0

1

1

2.5

2

4

3

5.5

4

7

5

8.5

f

f

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The graph below shows another way of solving an inequality. The horizontal

line y = 7 intersects the line y = 1.5x + 1 at the point (4, 7). Points on the line y = 1.5x + 1 above the horizontal line have yvalues greater than 7 and xvalues greater than 4. Thus, the solutions for 1.5x + 1 7 7 are x 7 4.

y 10

8

y = 7

6

4

y = 1.5x + 1

2

x O 2 4 6 8 10

The graph below shows that the equation y = 1.5x + 1 divides the coordinate plane into two regions. In the shaded region above the line, 1.5x + 1 6 y (or y 7 1.5x + 1), and in the unshaded region below the line, 1.5x + 1 7 y (or y 6 1.5x + 1).

y 8

6

1.5x + 1 < y

4 2

?2 O ?2

1.5x + 1 > y

x 2468

Properties of Inequality

The treatment of inequalities in this Unit is informal. Algebraic techniques for solving linear inequalities are covered in It's in the System.

The algebraic, numerical, and graphical strategies that lead to solutions of linear inequalities are related to those for equations, with some key differences. For example, you can multiply or divide both sides of an equation by a positive number without changing the solution. When you multiply or divide both sides of an inequality by a negative number, however, the direction of the inequality is reversed, as below.

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UNIT OVERVIEW

GOALS AND STANDARDS

MATHEMATICS BACKGROUND

UNIT INTRODUCTION

This is a true number sentence. 5 6 12

Multiply both sides of the inequality by - 1.

You do not get

-5 6 -12, which is a false

number sentence.

Instead, you get

-5 7 -12, which is another true

number sentence.

You may want to remind students of the properties of inequality.

Addition and Subtraction Properties of Inequality

Addition Property of Inequality

If you add the same number to each side of an inequality, the two sides remain equal.

Arithmetic

Algebra

8 < 12, so 8 + 3 < 12 + 3, and

10 > 7, so 10 + 5 > 7 + 5.

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

if a > b, then a + c > b + c.

Note: These relationships are also true for and .

Subtraction Property of Inequality

If you subtract the same number from each side of an inequality, the two sides remain equal.

Arithmetic

Algebra

8 < 12, so 8 - 4 < 12 - 4, and

10 > 7, so 10 - 2 > 7 - 2.

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

if a > b, then a - c > b - c.

Note: These relationships are also true for and .

In particular, make sure students understand the difference between the multiplication and division properties of inequality with positive numbers and the properties with negative numbers.

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