Linear Equations and Inequalities

Properties of Equality

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

Multiplication Property of Equality

If a = b, then a – c = b – c.

Addition Property of Equality

If a = b, then a × c = b × c.

Division Property of Equality If a = b, then a ÷ c = b ÷ c.

Subtraction Property of Equality

Let a, b, and c be real numbers.

Properties of Operations

a + b = b + a

Distributive Property

(a + b) + c = a + (b + c)

Commutative Property

a × (b + c) = (a × b) + (a × c)

Associative Property

a × b = b × a

(a × b) × c = a × (b × c)

a × (b – c) = (a × b) – (a × c)

Properties of Inequality

Multiplication Property of Inequality

Addition Property of Inequality

Division Property of Inequality

Subtraction Property of Inequality

If a > b, c > 0, then a ÷ c > b ÷ c. If a > b, c < 0, then a ÷ c < b ÷ c. If a < b, c < 0, then a ÷ c > b ÷ c. If a < b, c > 0, then a ÷ c < b ÷ c.

If a > b, c > 0, then a × c > b × c, If a > b, c < 0, then a × c < b × c. If a < b, c < 0, then a × c > b × c. If a < b, c > 0, then a ×

c < b × c.

What happens when you replace < with > or < or > in each of these properties? Does the property still hold? Try it!

Let a, b, and c be real numbers.

If a > b, then a + c > b + c. If a < b, then a + c < b + c.

If a > b, then a – c > b – c. If a < b, then a – c < b –

c.

Linear Equations with One Variable

An equation can be either a true statement or a false statement.

Which of these equations are true statements?

3(4 + 5) = 27

12 – 3 = 3 – 12

8 + 2 = 9

9 + 1 = 10

6 × 5 = 5 × 6

42 = 8

An equation with a variable can be either a true statement or a false statement, depending on the value given to the variable.

Which of these equations are true statements?

3(h + 5) = 27, h = 4 8 + x = 9, x = 1 6 × 5 = r × 6, r = 4

12 – 3 = 3 – m, m = –6y + 1 = 10, y = 8 42 = w, w = 8

Linear Equations with One Variable

An equation can be either a true statement or a false statement.

Which of these equations are true statements?

3(4 + 5) = 27

12 – 3 = 3 – 12

8 + 2 = 9

9 + 1 = 10

6 × 5 = 5 × 6

42 = 8

An equation with a variable can be either a true statement or a false statement, depending on the value given to the variable.

Which of these equations are true statements?

3(h + 5) = 27, h = 4 8 + x = 9, x = 1 6 × 5 = r × 6, r = 4

12 – 3 = 3 – m, m = –6y + 1 = 10, y = 8 42 = w, w = 8

Solving Linear Equations with One Variable

When solving an equation, you find the value(s) of the variable that makes the equation true.

Use your number sense to help you solve an equation with one variable.

y + 2 = 11think What number plus 2 equals 11?

nextSubstitute the number you have chosen into the original equation to check to see that it makes the equation true.

y + 2 = 11

9 + 2 = 11

11 = 11

Let y = 9.

Substitute 9 for y.

The equation is true for y = 9.

Solving Linear Equations with One Variable

Use properties to help you solve an equation with one variable.

y + 2 = 11

y + 2 – 2 = 11 – 2

y = 9

The solution checks.

Using the Subtraction Property of Equality, subtract 2 from both sides of the equation.

Check the solution.

y + 2 = 11

9 + 2 = 11

11 = 11

Check:

Solving Linear Inequalities with One Variable

x – 4 > 2

x – 4 + 4 > 2 + 4

x > 6

Using the Addition Property of Equality, add 4 to both sides of the equation.

Graph the solution on a number line.

x – 4 > 2

Use Properties to help you solve an inequality with one variable.

5 9-3 1 6 10-2 2-4 -1 0 3 4 7 8

x – 4 > 2

7 – 4 > 2

3 > 2

All values greater than 6 are solutions to this inequality.

The solution checks.

Try x = 7.

Choose a value greater than 6 and check it.

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