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1. One-step equations

Warm-up

Three quick questions from earlier lessons.

Discover

A balance has $x$ and 4 weights on one side and 11 weights on the other. What is $x$ ?

Key idea

Undo the operation on both sides.

An equation stays true if you do the same to both sides. Undo adding by subtracting, subtracting by adding, multiplying by dividing, dividing by multiplying.

Worked example

$5x = 35$ and $x - 8 = 12$ .

x = 7

Divide by 5.

x = 20

Add 8.

Your turn, guided

Your turn: $x + 9 = 23$ .

14

23 − 9.

Practice ★

Solve $x + 10 = 37$ . $x = \;?$
Solve $x + 23 = 35$ . $x = \;?$

Solve $11x = 77$ . $x = \;?$

Watch out!

Practice ★★

Solve $\dfrac{x}{ 3 } = 7$ . $x = \;?$
Solve $\dfrac{x}{ 3 } = 6$ . $x = \;?$

Exit ticket

Three questions to check you have got it.

Solve $x + 10 = 37$ . $x = \;?$
Solve $x + 23 = 35$ . $x = \;?$
Solve $x + 27 = 59$ . $x = \;?$

Summary card

One-step equations

Do the opposite operation to both sides.

Watch out: undo × with ÷ and ÷ with ×.

2. Solve linear inequalities in one variable

Warm-up

Three quick questions from earlier lessons.

Discover

$-2x > 6$ . Try $x = -4$ : $-2 \cdot (-4) = 8 > 6$ . So which describes all solutions?

$x < -3$
$x > -3$

Key idea

Solve inequalities like equations, with one extra rule.

Adding, subtracting, multiplying or dividing by a positive number keeps the inequality. Multiplying or dividing by a negative number flips it. Strict signs (

<

>

) exclude the endpoint;

\le

\ge

include it.

Worked example

Solve $-3x + 4 \le 13$ .

-3x \le 9

Subtract 4.

x \ge -3

Divide by −3: flip.

[-3, \infty)

Endpoint included.

Your turn, guided

Your turn: $2x - 1 < 9$ . The boundary value of $x$ .

x < 5

2x < 10.

Practice ★

Solve $8x + 5 < 45$ . Give the solution set as an interval.
Solve $6x + 5 < 59$ . Give the solution set as an interval.

You have 109 dollars and tickets cost 9 dollars each. Solve $9t \le 109$ : what is the greatest number of tickets you can buy?

Watch out!

Practice ★★

Solve $-2x + 17 \le 9$ . Give the solution set as an interval.
Solve $-2x + 1 \le 19$ . Give the solution set as an interval.

Exit ticket

Three questions to check you have got it.

Solve $8x + 5 < 45$ . Give the solution set as an interval.
Solve $6x + 5 < 59$ . Give the solution set as an interval.
Solve $7x + 9 < -54$ . Give the solution set as an interval.

Summary card

Inequalities

Solve as an equation; flip when multiplying or dividing by a negative.

Watch out: check with a number from your answer.

3. Equivalent expressions: distributing and combining like terms

Warm-up

Three quick questions from earlier lessons.

Discover

Is $10 - (3 + 2)$ the same as $10 - 3 + 2$ ?

Key idea

Equivalent expressions are equal for every value of the variable.

Distribute:

a(b + c) = ab + ac

, also with minus signs:

-(b - c) = -b + c

. Combine like terms by adding their coefficients, keeping each term's sign:

5x - 2x = 3x

. Think of subtraction as adding the opposite.

Worked example

Simplify $6x - 2(x - 4) + 1$ .

6x - 2x + 8 + 1

−2 times each term.

4x + 9

Combine.

Your turn, guided

Your turn: the coefficient of $x$ in $7x + 3 - 2x$ .

5

7 − 2.

Practice ★

Write $7(x - 3)$ without parentheses. What is the constant term?
Write $3(x - 7)$ without parentheses. What is the constant term?

Combine like terms in $9x + 6 - 11x + 4$ . What is the coefficient of $x$ ?

Watch out!

Practice ★★

Simplify $6x - (5x - 2)$ . What is the constant term?
Simplify $7x - (9x - 9)$ . What is the constant term?

Boss problem. A rectangle has sides $x + 6$ and $3x - 5$ . Its perimeter is $2(x + 6) + 2(3x - 5)$ .

Exit ticket

Three questions to check you have got it.

Write $7(x - 3)$ without parentheses. What is the constant term?
Write $3(x - 7)$ without parentheses. What is the constant term?
Write $3(x - 2)$ without parentheses. What is the constant term?

Summary card

Equivalent expressions

Distribute to every term; combine like terms with their signs.

Watch out: a minus before parentheses changes every sign inside.

4. Expressions with exponents; order of operations

Warm-up

Three quick questions from earlier lessons.

Discover

What is $3^4$ ?

Key idea

Exponents mean repeated multiplication; they come first.

b^n

means

n

factors of

b

b \cdot b \cdots b

. Order of operations: parentheses, then exponents, then multiplication and division, then addition and subtraction. In

3x^2

only

x

is squared.

Worked example

Evaluate $2 + 3 \cdot 4^2$ .

4^2 = 16

Exponent first.

3 \cdot 16 = 48

Multiply.

50

Add.

Your turn, guided

Your turn: $5x^2$ when $x = 3$ .

45

5 · 9.

Practice ★

What is the value of $6^{ 4 }$ ?
What is the value of $6^{ 2 }$ ?

Evaluate $1 + 2 \cdot 3^2$ .

Watch out!

Practice ★★

Evaluate $4x^2$ when $x = 8$ .
Evaluate $4x^2$ when $x = 4$ .

Boss problem. A square garden has side 10 m. A path 1 m wide goes all around it, making a bigger square.

Exit ticket

Three questions to check you have got it.

Evaluate $1 + 2 \cdot 3^2$ .
Evaluate $5 + 5 \cdot 2^2$ .
Evaluate $1 + 3 \cdot 2^2$ .

Summary card

Expressions with exponents

Parentheses, exponents, × ÷, + −.

Watch out: in kx² only x is squared.

5. Proportional relationships: y = kx

Warm-up

Three quick questions from earlier lessons.

Discover

3 pounds of apples cost 6 dollars. At the same price, what do 5 pounds cost?

10 dollars
8 dollars

Key idea

Proportional means $y = kx$ .

In a proportional relationship

y = kx

, where

k = \tfrac{y}{x}

is the constant of proportionality (the unit rate). Its graph is a line through the origin, and

k

is the slope.

Worked example

A printer prints 45 pages in 3 minutes.

k = 15

pages per minute

45 ÷ 3.

y = 15x

Equation.

120 pages in 8 minutes

x = 8.

Your turn, guided

Your turn: $y$ is proportional to $x$ , and $y = 28$ when $x = 4$ . Find $k$ .

7

28 ÷ 4.

Practice ★

$y$ is proportional to $x$ . When $x = 2$ , $y = 10$ . Find the constant of proportionality $k$ in $y = kx$ .
$y$ is proportional to $x$ . When $x = 3$ , $y = 15$ . Find the constant of proportionality $k$ in $y = kx$ .

Which equation represents a proportional relationship?
1. $y = 2x$
2. $y = 2x + 9$
3. $y = x + 2$

Watch out!

Practice ★★

A train travels 21 miles in 3 minutes at a constant speed. How far does it travel in 12 minutes?
A train travels 30 miles in 6 minutes at a constant speed. How far does it travel in 10 minutes?

Exit ticket

Three questions to check you have got it.

$y$ is proportional to $x$ . When $x = 2$ , $y = 10$ . Find the constant of proportionality $k$ in $y = kx$ .
$y$ is proportional to $x$ . When $x = 3$ , $y = 15$ . Find the constant of proportionality $k$ in $y = kx$ .
$y$ is proportional to $x$ . When $x = 4$ , $y = 28$ . Find the constant of proportionality $k$ in $y = kx$ .

Summary card

Proportional relationships

$y = kx$ , $k = y \div x$ ; the graph passes through $(0, 0)$ .

Watch out: scale by multiplying, not by adding.

Answers: Expressions and equations

1. One-step equations

$27$
$12$

$21$
$18$

$27$
$12$
$32$

2. Solve linear inequalities in one variable

$(-\infty, 5)$
$(-\infty, 9)$

$12$

$[4, \infty)$
$[-9, \infty)$

$(-\infty, 5)$
$(-\infty, 9)$
$(-\infty, -9)$

3. Equivalent expressions: distributing and combining like terms

$-21$
$-21$

$-2$

$-21$
$-21$
$-6$

4. Expressions with exponents; order of operations

$1296$
$36$

$19$

$256$
$64$

$44$

$19$
$25$
$13$

5. Proportional relationships: y = kx

$y = 2x$

$84$
$50$