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1. 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.

2. Solve a formula for a variable

Warm-up

Three quick questions from earlier lessons.

Discover

A rectangle has perimeter $P = 2\ell + 2w$ . You know $P$ and $\ell$ . Which formula gives $w$ directly?

$w = \dfrac{P - 2\ell}{2}$
$w = P - \dfrac{2\ell}{2}$

Key idea

Solve for a letter as you solve for $x$ .

To solve for a variable, treat the other letters as numbers: use the same moves as for a number equation until that variable is alone. Divide every term on the other side.

Worked example

Solve $3x + 2y = 12$ for $y$ .

2y = 12 - 3x

Subtract 3x.

y = \dfrac{12 - 3x}{2}

Divide both terms by 2.

Your turn, guided

Your turn: $y = 4x + 8$ . Solve for $x$ .

x = \dfrac{y - 8}{4}

Subtract 8, divide by 4.

Practice ★

A car moves at 2 miles per minute, so $d = 2t$ . Solve for $t$ . $t = \;?$
A car moves at 4 miles per minute, so $d = 4t$ . Solve for $t$ . $t = \;?$

Solve $7x + 9y = 23$ for $y$ . $y = \;?$

Watch out!

Practice ★★

Solve $y = 9x + 3$ for $x$ . $x = \;?$
Solve $y = 2x + 4$ for $x$ . $x = \;?$

Exit ticket

Three questions to check you have got it.

Solve $7x + 9y = 23$ for $y$ . $y = \;?$
Solve $7x + 5y = 13$ for $y$ . $y = \;?$
Solve $7x + 8y = 7$ for $y$ . $y = \;?$

Summary card

Solving for a variable

Same moves as with numbers; the other letters are just numbers you do not know.

Watch out: divide 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. Absolute value equations and inequalities

Warm-up

Three quick questions from earlier lessons.

Discover

Which numbers are at a distance of 3 from 0 on the number line?

3 and −3
Only 3

Key idea

Absolute value is distance.

|x - a|

is the distance between

x

and

a

. So

|x - a| = b

(with

b > 0

) has two solutions,

a - b

and

a + b

;

|x - a| < b

is the interval

a - b < x < a + b

;

|x - a| > b

is the two outer pieces

x < a - b

x > a + b

. If

b < 0

|x - a| = b

has no solution.

Worked example

Solve $|x - 2| = 5$ and $|x - 2| \le 5$ .

x - 2 = 5

x - 2 = -5

x = 7

x = -3

Two cases.

-3 \le x \le 7

Inequality.

Your turn, guided

Your turn: the larger solution of $|x - 4| = 6$ .

10

4 + 6.

Practice ★

Solve $|x - (4)| = 4$ . Enter all solutions.
Solve $|x - (-6)| = 5$ . Enter all solutions.

How many solutions does the equation |x + 9| = −7 have?
1. None
2. Two
3. One: x = -2

Watch out!

Practice ★★

Solve $|x - (4)| < 4$ . Give the solution set as an interval.
Solve $|x - (9)| < 6$ . Give the solution set as an interval.

Solve $|x - (2)| \geq 2$ . Give the solution set as an interval or a union of intervals.

Exit ticket

Three questions to check you have got it.

Solve $|x - (4)| = 4$ . Enter all solutions.
Solve $|x - (-6)| = 5$ . Enter all solutions.
Solve $|x - (4)| = 3$ . Enter all solutions.

Summary card

Absolute value equations and inequalities

|x − a| = b: a ± b. Less than: between. Greater than: outside.

Watch out: do not forget the negative case.

5. Linear equations in two variables and their graphs

Warm-up

Three quick questions from earlier lessons.

Discover

Is $(2, 4)$ a solution of $3x + y = 10$ ?

Key idea

A linear equation in two variables is a line.

The solutions of

Ax + By = C

are the points of a line. Set

y = 0

for the $x$ -intercept and

x = 0

for the $y$ -intercept. Solving for

y

gives

y = mx + b

, with slope

m = -\tfrac{A}{B}

Worked example

$2x + 5y = 20$ .

x

-intercept

10

y = 0.

y

-intercept

4

x = 0.

Slope

-\tfrac{2}{5}

y = −(2/5)x + 4.

Your turn, guided

Your turn: the $y$ -intercept of $3x + 4y = 24$ .

6

x = 0: 4y = 24.

Practice ★

The point $(2, y)$ is a solution of $5x + y = 15$ . Find $y$ .
The point $(1, y)$ is a solution of $5x + y = 17$ . Find $y$ .

Find the $x$ -intercept of $3x + 6y = 54$ (the value of $x$ where the line crosses the $x$ -axis).

Watch out!

Practice ★★

What is the slope of the line $8x + 9y = 17$ ?
What is the slope of the line $1x + 4y = 4$ ?

Exit ticket

Three questions to check you have got it.

Find the $x$ -intercept of $3x + 6y = 54$ (the value of $x$ where the line crosses the $x$ -axis).
Find the $x$ -intercept of $3x + 1y = 3$ (the value of $x$ where the line crosses the $x$ -axis).
Find the $x$ -intercept of $4x + 1y = 12$ (the value of $x$ where the line crosses the $x$ -axis).

Summary card

Lines from equations

Intercepts: set the other variable to 0. Slope: solve for $y$ .

Watch out: slope of $Ax + By = C$ is $-A/B$ .

Answers: Linear equations and inequalities

1. Solve linear equations in one variable

$-7$
$-5$

$-7$
$-5$
$7$

2. Solve a formula for a variable

$\frac{d}{2}$
$\frac{d}{4}$

$\frac{23}{9} - \frac{7 x}{9}$

$\frac{y}{9} - \frac{1}{3}$
$\frac{y}{2} - 2$

$\frac{23}{9} - \frac{7 x}{9}$
$\frac{13}{5} - \frac{7 x}{5}$
$\frac{7}{8} - \frac{7 x}{8}$

3. Solve linear inequalities in one variable

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

$12$

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

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

4. Absolute value equations and inequalities

$0,\ 8$
$-11,\ -1$

None

$(0, 8)$
$(3, 15)$

$(-\infty, 0] \cup [4, \infty)$

$0,\ 8$
$-11,\ -1$
$1,\ 7$

5. Linear equations in two variables and their graphs

$5$
$12$

$18$

$- \frac{8}{9}$
$- \frac{1}{4}$

$18$
$1$
$3$