Download 1E Solving Equations and Inequalities

1E Equations and Inequalities
Solving Equations
1.
2.
3.
Simplify each side by adding/subtracting
Move the variables to one side and the numbers to the other side by doing the OPPOSITE.
Multiply by what is below ‘x’ and divide by what is next to ‘x’.
Example 1: 2(x + 3) = 8 Ex. 2: 2x – 5x + 6 = 8 + x Ex 3: 3 x + 1 = -11
Example 4: 3 – (x + 2) = 8
4
2x + 6 = 8
2x
2x
x
x
=8–6
-3x + 6 = 8 + x
3
x
4
= -11 – 1
3–x–2
=8
=8–6
3
x
4
= -12
1–x
=8
-x -3x
=2
= 2/2
=1
-4x= 2
x = 2/-4
x = -1/2
x
x
x
= -12*4/3
= -48/3
= -16
Practice. 1-15: Solve
1) 5x + 2 = 27
2) 5x + 16 = 51
3) 8 = -12 + 5k
4) 6- x = 5x + 30
6) -4 – 3x = 6x – 6
7) 18 – 4k = 2 + 5 – 3k
8) 20 – a = 10a -2
9) ½x + 3 = -5
11) 18 = 3(2c + 2)
12) 3(-2 – 3x) = -9x – 4
13) ½(4g + 2) = 8
14) 6 – (g + 3) = 8g
15) Three less than a number is 8.
4
-x
-x
x
=8-1
=7
= 7/-1 = -7
5) 4x – 8 = 10 – 2x
10) (3/5)x + 3 = 24
16) Twice the sum of a number and eight is 16
17) James weighs 35 pounds less than twice Sarah’s weight. James weighs 165. How much does Sarah weigh?
18) The amount of dogs is three-fourths the amount of cats. There are 27 dogs. How many total animals are there?
19) The length of a rectangle is 2x + 1 and the width is 3x – 2. The perimeter is 48. Find x and the length
20) The three sides of a triangle are 5x + 1; 2x – 3 and 7x + 8. The perimeter is 48. Find the longest side.
21) The cost to download songs is C=2S + 4. Jack downloads 10 songs and has $15. How much more money does he need?
22) The cost of rides at a fair is C=3R + 7. Jack wants to ride 8 rides. He has $20. How much more money does he need?
1E Part 2: Inequalities Investigation
Graph the inequalities.
2. m ≥-2
1. x > 4
3. - 1 ≥a
When you graph, when do we use an open dot?
_____________________________________________
When you graph, when do we use a closed dot?
_____________________________________________
Consider the following true statements:
Multiply each number by -1,
and then fill in the inequality symbol.
3<7
-2 < 1
-8 < -4
(-1)(3) ___ (-1)(7)
_____
_____
_____
_____
>
_____
_____
_____
_____
_-3_
Consider the following true statements:
Divide each number by -2,
and then fill in the inequality symbol.
_-7_
4>2
8 > -4
-6 > -10
4
2
 2 ___  2
_____
_____
_____
_____
<
_____
_____
_____
_____
_-2_
_-1_
What can you conclude about multiplying or dividing by a negative number? ________________________________
SUMMARY: The inequality symbol changes direction (ONLY) when dividing or multiplying by a
negative number.
For graphing: < or > use an open circle
Solve and graph these inequalities
4. 4x + 3 > 23
5. 3x – 10 ≤ 26
8.
6-
3
x < 15
4
≤ or ≥ use a closed circle
6. 12 – 7x > 54
7. 40 – 5x ≥ - 10
1E Part 3: Solving Inequalities.
Same as solving equalities except that you change the inequality (<, >, <, >) when you divide or multiply by a negative number At
<
Less than
Fewer than
Ex 5: 3x - 2 > -14
3x
> -14 + 2
3x
> -12
x
> -12/3
x
> -4
<
Less than or equal to
At most
No more than
A maximum of
Ex 6: 2 – x > 8
-x > 8 – 2
-x > 6
x < 6/-1 change > to <
x < -6
Practice 23-33: Solve.
23) 4 + 2x > - 12
24) 7 – 3x > -14
25) 19 – ½ x > 7
26) 16 – 3x < 22
27) 3( 2 – x) > 8
28) 7(5 + x) < -6
29) 5 + 2(x – 3) > 7 – x
30) 3 – 5x + 2x > 3
31) Three times a number is at least 12
>
Greater than
More than
>
Greater than or equal to
At least
No less than
A minimum of
Ex 7: 5 + 3x > 7x – 6
x/-2 < 6
6 + 5 > 7x – 3x
x
> 6 * -2 change
11
> 4x
x > -12
11/4 > x
32) The sum of Tom’s age and Sally’s age is at most 28. Tom is 12.
33) The length of rectangle is 5x – 3.The width is 2x + 1.The perimeter is at most 50. What is the largest possible value for x if x is an
integer?
1E Part 4: Word Problems on Linear Equations and Inequalities
34. Jackson has $15 in his “Digital Download Account.” Each song he downloads costs $4. How much money does he need to
put into his “Digital Download Account” so that he can download 9 songs?
35. Julie has $19 in her wallet. Each pretzel costs $5. What is the maximum amount of pretzels she can buy? How much will
she have left over?
36. A box of 50 lollipops costs $10. A group wants to make a profit of $4 per box. How much do they need to sell each lollipop
for?
37. A box 36 candy bars costs $18. A group wants to make a profit of $9 per box. How much do they need to sell each candy
bar for?
38. Burritos cost 5$ each and Tacos cost 3$ each.
If Jack has $30 to spend, write an inequality that shows how many tacos he
can buy? (Let B represent burritos and T represent tacos.)
39. Cat toys are $2 each and Dog toys are $3 each. Tom has $18 to spend. Let x represent the cat toys and y represent the dog
toys. Write an inequality that represents how much Tom can spend on dog toys.
40. Twice the cost of apples and five times the amount of oranges is at most $24. Let x represent the cost of apples and y
represent the cost of oranges. Write an inequality that represents the cost of oranges in terms of apples.
41. Let x represent the cost of candy bars and y represent the cost of sodas. Write an inequality for each of the following
situations in terms of the soda. Then match the correct one to the equation Y < 2x – 4:
a. Four times the candy bars and two times the soda is at least 8.
b. The difference of four times the candy bars and twice the sodas is at most 8.
c.
The difference of four times the candy bars and twice the sodas is more than 8.
42. Jack makes $50 a day plus 5% commission on the price of cars he sells. How much does he have to sell if he wants to make
$200?
43. Mark has $13 in his wallet. Each video game costs $4. He wants to rent 6 video games. How much more money does he
need?
44. A box of 50 spirit pompoms cost $25. If the cheerleaders want to make $10 per box of pompoms, how much do they have to
sell each pompom for?
45. True-false questions on a test are 3 points each and multiple-choice questions are 5 points each. Let x represent True-false
and y represent the amount of multiple-choice. If Jake wants to get at least 75 points on the test, write an inequality in terms
of multiple-choice questions.
46. Lisa makes $20 in wages plus 15% in tips. How much does she have serve in order to make at least $170?
47. The difference of 4 times the price of a children’s ticket and 3 times the price of an adult ticket is at least $24. Let x represent
children’s ticket and y represent adult’s ticket. Write an inequality that represents the adult’s ticket.