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1. 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.

2. Solve linear equations in one variable

Warm-up

Three quick questions from earlier lessons.

Discover

A pizza party costs 40 dollars plus 12 dollars per pizza, and the total is 100 dollars. Which equation fits?

$40 + 12p = 100$
$12 + 40p = 100$

Key idea

Equivalent equations have the same solutions.

To solve a linear equation, make equivalent equations: add or subtract the same amount on both sides, multiply or divide both sides by the same non-zero number, distribute, collect like terms. Stop when

x

is alone.

Worked example

Solve $5x + 3 = 2x + 15$ .

3x + 3 = 15

Subtract 2x.

3x = 12

Subtract 3.

x = 4

Divide by 3.

Your turn, guided

Your turn: $3(x + 2) = 21$ . Find $x$ .

x = 5

Divide by 3: x + 2 = 7.

Practice ★

Solve $3x + 3 = -18$ . $x = \;?$
Solve $8x + 14 = -26$ . $x = \;?$

Solve $4x + 8 = 3x + 9$ . $x = \;?$

Watch out!

Practice ★★

Solve $6(x + 4) = 30$ . $x = \;?$
Solve $3(x + 4) = 33$ . $x = \;?$

Exit ticket

Three questions to check you have got it.

Solve $3x + 3 = -18$ . $x = \;?$
Solve $8x + 14 = -26$ . $x = \;?$
Solve $4x + 4 = 32$ . $x = \;?$

Summary card

Linear equations

Do the same to both sides until $x$ is alone.

Watch out: distribute to every term.

3. 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.

4. Arithmetic sequences

Warm-up

Three quick questions from earlier lessons.

Discover

A taxi charges 3 dollars, plus 2 dollars per mile. Mile by mile, the fare is $5, 7, 9, \dots$ What does the 10th mile cost in total?

23 dollars
25 dollars

Key idea

An arithmetic sequence adds the same number $d$ (the common difference) each time.

a_n = a_1 + (n - 1)d

. From

a_p

a_q

there are

q - p

steps, so

d = \dfrac{a_q - a_p}{q - p}

Worked example

An arithmetic sequence has $a_3 = 11$ and $a_8 = 31$ . Find $d$ and $a_{20}$ .

d = \dfrac{31 - 11}{5} = 4

5 steps from a₃ to a₈.

a_1 = 11 - 2 \cdot 4 = 3

Back 2 steps from a₃.

a_{20} = 3 + 19 \cdot 4 = 79

19 steps from a₁.

Your turn, guided

Your turn: $a_1 = 7$ , $d = 5$ . Find $a_{12}$ .

a_{12} = 7 + 11 \cdot 5

11 steps.

a_{12} = 62

7 + 55.

Practice ★

An arithmetic sequence has $a_1 = 14$ and common difference 2. Find $a_{ 43 }$ .
An arithmetic sequence has $a_1 = 17$ and common difference 3. Find $a_{ 24 }$ .
An arithmetic sequence has $a_1 = 8$ and common difference 5. Find $a_{ 26 }$ .
An arithmetic sequence has $a_1 = 20$ and common difference 5. Find $a_{ 28 }$ .

Watch out!

Practice ★★

In an arithmetic sequence, $a_{ 4 } = 28$ and $a_{ 9 } = 68$ . Find the common difference.
In an arithmetic sequence, $a_{ 3 } = 18$ and $a_{ 7 } = 34$ . Find the common difference.
In an arithmetic sequence, $a_{ 3 } = 11$ and $a_{ 10 } = 25$ . Find the common difference.

What kind of sequence is $3, 6, 12, 24, \dots$ ?
1. Geometric, with ratio 2
2. Arithmetic
3. Neither

Exit ticket

Three questions to check you have got it.

An arithmetic sequence has $a_1 = 14$ and common difference 2. Find $a_{ 43 }$ .
An arithmetic sequence has $a_1 = 17$ and common difference 3. Find $a_{ 24 }$ .
An arithmetic sequence has $a_1 = 8$ and common difference 5. Find $a_{ 26 }$ .

Summary card

Arithmetic sequences

Add $d$ each time: $a_n = a_1 + (n - 1)d$ .
$d = \dfrac{a_q - a_p}{q - p}$ .

Watch out: $n - 1$ steps from $a_1$ to $a_n$ .

5. Geometric sequences

Warm-up

Three quick questions from earlier lessons.

Discover

A rumor spreads: 3 people know it on day 1, and each day the number triples. How many know it on day 4?

Key idea

A geometric sequence multiplies by the same number $r$ (the common ratio) each time.

a_n = a_1 \cdot r^{\,n-1}

, and

r = \dfrac{a_{n+1}}{a_n}

Worked example

Find $a_7$ for $2, 6, 18, \dots$

r = 3

6 ÷ 2.

a_7 = 2 \cdot 3^6

6 multiplications from a₁ to a₇.

a_7 = 1458

3⁶ = 729.

Your turn, guided

Your turn: $a_1 = 5$ , $r = 2$ . Find $a_6$ .

a_6 = 5 \cdot 2^5

5 multiplications.

a_6 = 160

5 · 32.

Practice ★

A geometric sequence has $a_1 = 3$ and ratio 3. Find $a_{ 4 }$ .
A geometric sequence has $a_1 = 1$ and ratio 2. Find $a_{ 8 }$ .
A geometric sequence has $a_1 = 2$ and ratio 3. Find $a_{ 8 }$ .
A geometric sequence has $a_1 = 4$ and ratio 2. Find $a_{ 5 }$ .

Watch out!

Practice ★★

Find the common ratio of the geometric sequence $8, 32, 128, \dots$
Find the common ratio of the geometric sequence $7, 21, 63, \dots$

Write a formula for $a_n$ , starting at $n = 1$ : $2, 8, 32, \dots$
Write a formula for $a_n$ , starting at $n = 1$ : $9, 36, 144, \dots$
Write a formula for $a_n$ , starting at $n = 1$ : $8, 40, 200, \dots$

Exit ticket

Three questions to check you have got it.

Write a formula for $a_n$ , starting at $n = 1$ : $2, 8, 32, \dots$
Write a formula for $a_n$ , starting at $n = 1$ : $9, 36, 144, \dots$
Write a formula for $a_n$ , starting at $n = 1$ : $8, 40, 200, \dots$

Summary card

Geometric sequences

Multiply by $r$ each time: $a_n = a_1 r^{\,n-1}$ .
$r$ = a term divided by the one before.

Watch out: the exponent is $n - 1$ .

Answers: Expressions, equations and inequalities

1. Equivalent expressions: distributing and combining like terms

$-21$
$-21$

$-2$

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

2. Solve linear equations in one variable

$-7$
$-5$

$-7$
$-5$
$7$

3. Solve linear inequalities in one variable

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

$12$

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

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

4. Arithmetic sequences

$98$
$86$
$133$
$155$

Geometric, with ratio 2

$98$
$86$
$133$

5. Geometric sequences

$81$
$128$
$4374$
$64$

$2 \cdot 4^{n - 1}$
$9 \cdot 4^{n - 1}$
$8 \cdot 5^{n - 1}$

$2 \cdot 4^{n - 1}$
$9 \cdot 4^{n - 1}$
$8 \cdot 5^{n - 1}$