Unit 2B-Equations and Inequalities

1 step equations

100

Show all work: x + 4 = 10

What is

x+4 - 4= 10 - 4

x = 6

200

Solve. Show all work: x - ⅔ = 4

What is

x - ⅔ = 4

x - ⅔ + ⅔ = 4 + ⅔

x = 4 ⅔

300

Solve. Show all work.

⅕ x=3

What is

⅕x=3

5 ⅕ x=3*5

x=15

400

Solve. Show all work.

1.2x = 2.4

What is

1.2x = 2.4

^1.2/_1.2 x = ^2.4/_1.2

x = 2

500

Solve. Show all work.

¾x = 6

What is

¾x = 6

4* ¾ x = 6*4

3x = 24

³/₃x = ²⁴/₆

= 4

2-step equations

100

Solve. Show all work. 8x - 7 = 17

What is

8x-7+7=17+7

⁸/₈x=²⁴/₈

x=3

200

#2 Solve. Show all work.

4 - 6x = 8

4-4 - 6x = 8-4

^{- 6}/_-6 x = ⁴/_-6

x = ^-4/₆ = ^-2/₃

300

#3 Solve. Show all work.

6 - ^y/₃ = 0

6-6 - ^y/₃ = 0-6

- ^y/₃ = -6

- ^y/₃*-3 = -6*-3

y = 18

400

#4 Solve. Show all work.

3 - 3.6x = 4.2

3*10 - 3.6*10x = 4.2*10

30 - 36x = 42

30-30 - 36x = 42-30

- 36x = 12

^-36/_-36x = ¹²/_-36

x = ^-12/₃₆=^-12/12/_36/12=^-1/₃

500

#5 Solve. Show all work. 7x - 5 = 3x + 4

7x-3x - 5 = 3x-3x + 4

4x - 5+5 = 4+5

4x = 9

⁴/₄x = ⁹/₄

x= ⁹/₄

Draw Solutions on a Number Line

100

x < -4

Make sure to clearly mark where your circle is on the number line and which direction you are shading

The solution has an open circle at -4 and is shaded to the left.

200

x > 10

Make sure to clearly mark where your circle is on the number line and which direction you are shading

The solution has a solid circle at 10 and is shaded to the right.

300

x < 9

Make sure to clearly mark where your circle is on the number line and which direction you are shading

The solution has an solid circle at 9 and is shaded to the left.

400

x < ⅓

Make sure to clearly mark where your circle is on the number line, which whole numbers it is in between and which direction you are shading

The solution has an open circle at ⅓ which is in between 0 and 1 and is shaded to the left.

500

-3 < x

Make sure to clearly mark where your circle is on the number line and which direction you are shading

Switch the inequality to: x >-3 The solution has an open circle at -3 and is shaded to the right.

Word Problems

100

#23 The sum of the page numbers of two facing pages in a book is 145. What are the page numbers?

Two facing pages are 2 numbers right after each other like 2 and 3. So if we say the first page number is x, the second page number is x+1. When we add the page numbers together it is equal to 145.

x + (x+1) = 145

2x+1 - 1= 145 - 1

²/₂ x = ¹⁴⁴/₂

x=72, so the pages are 72 and 73

200

#25 The sum of the interior angle measures of a quadrilateral is 360. The measure of angle A is three times the measure of angle D. The measure of angle B is four times that of angle D. The measure of angle C is 24 more than angle B. Find the measure of each angle of the quadrilateral.

The facts from annotating the problem are: angle D=x, angle A = 3x, angle B=4x, and angle C=24.

x+ 3x+4x+24 = 360

8x + 24 = 360 Add like terms

8x + 24 - 24 = 360 - 24 Inverse addition and DSBS

^{8/₈ x = 336/₈ Inverse * and DSBS}

x = 42, so angle D = 42, Angle A=3*42=126, Angle B=4*42=168

checking: 42+126+168+24=360

300

#30 The coach of the field hockey team can spend at most $475 on new team uniforms. The coach will order the uniforms online and pay a mailing cost of $6.50. If each uniform costs $29, how many uniforms can the coach order?

29u + 6.50 < 475

29u + 6.50 - 6.50 < 475-6.50

²⁹/₂₉u < ⁴⁷⁵/₂₉

u < 16.4 SO the coach can order 16 uniforms

400

#26 Laura want the average amount of money she spends each day on her four-day vacation to be no more than $64. On the first three days, she spends $71, $62, and $59. What is the greatest amount of money she can spend on the fourth day?

^(71+62+59+x)/₄ < 64 Add the 4 days of spending, including the fourth day which is x, the unknown.

4* ^(71+62+59+x)/₄ < 4* 64

(71+62+59+x) < 256

192-192 +x < 256-192

x < 64 So she can spend up to $64 on day four

500

Kevin plans to sign up for p hours of training at a culinary school. The school offers two payment options as shown.

Option A: $8 per hour plus $200 for supplies Option B: $18 per hour, no fee for supplies

For how many hours of training is Option B less expensive than Option A?

18h<8h+200

18h - 8h < 8h - 8h + 200

¹⁰/₁₀ < ²⁰⁰/₁₀

h < 20 So option B is less expensive when you go for less than 20 hours

Inequalities

100

#11 Solve. Show all work. Graph each solution set .

4x - 3 > 1

4x - 3+3 > 1+3

4x > 4

⁴/₄x >⁴/₄

x >1

The number line has an open circle at 1 and is shaded to the right.

200

#12 Solve. Show all work. Graph the solution set.

6 < 1 - 5x

6 -1< 1-1 - 5x

5< - 5x

⁵/_-5< ^-5/_-5x

-1 > x

OR x < -1

(The symbol switches because you are * or / by a negative number

The number line has a solid circle at -1 and is shaded to the left.

300

#13 Solve. Show all work. Graph the solution ⅔ - ^x/₆ > -½

⅔ - ^x/₆ > -&frac12

6*⅔ - 6*^x/₆ > 6*-½

4- x > -3

4- 4 - x > -3-4

^-1/_-1 x > ^-7/_-1

x < 7 The sign switches direction because you are dividing by a negative

The number line has a shaded circle at 7 and is shaded to the left.

400

#15 Solve. Show all work. Graph the solution 9y-5 < 4y+15

9y-4y -5 < 4y-4y+15

5y -5+5 < 15+5

5y < 20

⁵/₅y < ²⁰/₅

y < 4

The number line has a solid circle at 4 and is shaded to the left.

500

#18 Solve. Show all work. Graph the solution 3(x+1) > 5x+7

3x+3 > 5x+7 use the distributive property

3x-3x+3 > 5x-3x+7 balance variable term

3-7 > 2x+7-7 balance constant term

-4 > 2x

^-4/₂ > ²/₂x

-2 > x OR x < -2 flip so the variable is on the left side

The number line has an open circle at -2 and is shaded to the left.