Transcript Solving Equations - Sr. Fatima's Class

Solving Equations and
Inequalities
Vocabulary
An equation is a sentence stating two quantities are equal.
An algebraic equation is an equation that includes one or more
variables.
An equivalent equations is an equation that have the same
solution(s). For example: x + 2 = 6, if x = 4 both sides of the
equation are equal.
Isolate – to solve an equation you must isolate the variable (i.e.
get the variable alone to one side of the equation).
An inverse operation undoes another operation by performing the
opposite operation (i.e subtraction is the inverse operation of
addition, multiplication is the inverse operation of division).
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Properties of Equality –
producing equivalent equations
Addition Property of Equality: Adding the same number to
each side of an equation produces an equivalent equation.
x–3=2
x–3+3=2+3
Subtraction Property of Equality: Subtracting the same
number to each side of an equation produces an
equivalent equation.
x+3=2
x+3-3=2-3
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Properties of Equality –
producing equivalent equations
Multiplication Property of Equality: Adding the same number to
each side of an equation produces an equivalent equation.
𝑥
3
𝑥
3
=2
∙3=2∙3
Division Property of Equality: Dividing the same number to each
side of an equation produces an equivalent equation.
5x = 20
5𝑥
𝟓
=
20
𝟓
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Solving One-Step Equations
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Solving an One-Step Equation
Solving an equation with subtraction
Solving an equation with addition
x + 13 = 27
-7 = b - 3
isolate the variable
-7 + 3 = b – 3 + 3
undo subtraction by
adding the same number
from each side
Addition Property of
Equality
-4 = b
Simplify each side of the
equation
isolate the variable
x + 13 – 13 = 27 – 13 undo addition by
subtracting the same
number from each side
Subtraction Property of
Equality
x = 14
Simplify each side of the
equation
Check it!
x + 13 = 27
14 + 13 = 27
27 = 27
Check it!
-7 = b - 3
Substitute the answer into
original equation to check
it.
-7 = -4 - 3
-7 = -7
Substitute the answer into
original equation to check
it.
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Solving an One-Step Equation
Solving an equation with
multiplication
4x = 28
isolate the variable
4𝑥
4
undo multiplication by
dividing the same number
from each side
Division Property of
Equality
=
28
4
x=7
Simplify each side of the
equation
Check it!
4x = 28
4 ∙ 7 = 28
28 = 28
Substitute the answer into
original equation to check
it.
Solving an equation with division
𝒙
𝟒
𝒙
𝟒
= -9
isolate the variable
∙ 𝟒 = -9 ∙ 4
undo division by
multiplying the same
number from each side
Multiplication Property of
Equality
x = -36
Check it!
𝒙
= -9
𝟒
−𝟑𝟔
= -9
𝟒
-9 = -9
Simplify each side of the
equation
Substitute the answer into
original equation to check
it.
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Solving an One-Step Equation
solve equations with rational coefficients
Solving an equation using reciprocal aka multiplicative inverses
Two numbers with a product of 1 are called multiplicative inverses or reciprocals.
Inverse Property of Multiplication
The product of a number and its multiplicative inverse is 1.
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𝑎 𝑏
Example: × = 1
∙ = 1, a,b ≠ 0
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𝟒
𝒎
𝟓
3
𝑏
𝑎
isolate the variable
= 28
𝟓 𝟒
5
𝒎 = (28)
𝟒 𝟓
4
m = 35
Check it!
Multiply each side by 5/4, the
reciprocal of 4/5
Inverse Property of Multiplication
Simplify each side of the equation
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Solving Two and Multi -Step
Equations
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Understanding
Two-Step
To solve two-step equations, you use the properties of
equality and inverse operations to form a series of simpler
equivalent equations. You can use the properties of equality
repeatedly to isolate the variable.
Multi-Step
To solve two-step equations, you use the properties of
equality, inverse operations, and properties of real numbers
to form a series of simpler equivalent equations. You use the
properties until you isolate the variable.
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You can undo the operations in the reverse
order of the order of operations.
2x + 3 = 15
2x + 3 – 3 = 15 – 3
subtract 3 from each side
2x = 12
simplify
2𝑥
2
Divide each side by 2
=
12
2
x=6
Simplify
Check
2x + 3 = 15
2(6) + 3 = 15
15 = 15
Solving Two-Step Equations
Step 1: Undo the addition or subtraction first
Step 2: Then undo the multiplication or division
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Solving Equations with Variables
on Both Sides
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How To Get Started
There are variables terms on both sides of the equation.
Decide which variable term to add or subtract to get the
variable on one side only.
To solve equations with variables on both sides, you can
use the properties of equality and inverse operations to
write a series of simpler equivalent equations.
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Solving an Equation with Variables on Both Sides
5x = 2x + 15
Original equation
5x – 2x = 2x +15 – 2x
Subtract 2x from each side
Subtraction Property of Equality
3x = 15
Simplify
𝟑𝒙 𝟏𝟓
=
𝟑
𝟑
Divide each side by 3
Division Property of Equality
x=5
Simplify
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Solving Inequalities
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What is an inequality
An inequality is a mathematical sentence that compares quantities
Words
Symbols
x is less than two
x<2
x is greater than or equal to four
x≥4
Inequalities
Words
Symbols
• is less than
• is greater than • is less than or equal to • is greater than or equal to
• Is fewer than • is more than
• is no more than
• is no less than
• exceeds
• is at most
• is at least
<
>
≤
≥
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Addition and Subtraction
Properties of Inequality
Addition and Subtraction Properties of Inequality
When you add or subtract the same number from each side of an
inequality, the inequality remains the same.
For all numbers a, b, and c,
1. If a > b, then a + c > b + c and a – c > b – c
2. If a < b, then a + c < b + c and a – c < b – c
Example
2<4
6>3
2+3<4+3
6–4>3–4
5<7
2>1
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Solve Inequalities using Addition and Subtraction
x + 3 > 10
Original equation
-6 ≥ n – 5
Original equation
x + 3 – 3 > 10 – 3
Subtract 3 from each
side
-6 + 5 ≥ n – 5 + 5
Add 5 to both
sides
x>7
Simplify
-1 ≥ n
Simplify
Therefore, the solution is x > 7
Therefore, the solution is -1 ≥ n or n ≤ -1
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Multiplication and Division
Properties of Inequality, Positive Numbers
Multiplication and Division Properties of Inequality,
Positive Number
When you multiply or divide each side of an inequality by a positive
number, the inequality remains the same.
For all numbers a, b, and c,
where c > 0
𝑎
𝑏
>
𝑐
𝑐
𝑎
𝑏
1. If a < b, then ac < bc and <
𝑐
𝑐
1. If a > b, then ac > bc and
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Solve Inequalities using Multiplication and Division,
positive numbers
8x ≤ 40
Original equation
𝒅
𝟐
8𝑥 40
≤
8
8
Divide each side by 8
2
x≤5
Simplify
d > 14
The solution is x ≤ 5. You can check this
solution by substituting 5 or a number less than
5 into the inequality
Original equation
>7
𝑑
2
> 2(7)
Multiply both sides
by 2
The solution is d > 14. You can check this
solution by substituting a number greater
than 14 into the inequality
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Multiplication and Division
Properties of Inequality, Negative Numbers
Multiplication and Division Properties of Inequality,
Negative Number
When you multiply or divide each side of an inequality by a negative
number, the inequality symbol must be reversed for the inequality to
remains true.
For all numbers a, b, and c,
where c < 0
𝑎
𝑐
𝑎
1. If a < b, then ac > bc and
𝑐
7>1
-2(7) < -2(1)
REVERSE THE SYMBOLS
-14 < -2
1. If a > b, then ac < bc and
Example
<
>
𝑏
𝑐
𝑏
𝑐
-4 < 16
−4 16
>
−4 −4
1 > -4
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Solve Inequalities using Multiplication and Division,
negative numbers
𝒙
≤𝟒
−𝟑
−3
𝑥
−3
x ≥ -12
≥ -3(4)
Original equation
Multiply each side
by -3 and reverse the
symbol
Simplify
The solution is x ≥ -12. You can check this
solution by substituting -12 or a number
greater than -12 into the inequality
-6a ≥ -78
−6𝑎 −78
≤
−6
−6
Original equation
Divide both sides by
-6 and reverse the
symbol
a ≤ 13
The solution is a ≤ 13. You can check this
solution by substituting 13 or a number less
than 13 into the inequality
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Check Your UNDERSTANDING
Solve each equation. Check your answer.
Solving one-step equations
Solving one-step equations
1. n + 6 = 8
6. 6c = 18
2. -2 = a + 6
7. -8x = 24
3. X - 5 = 6
8.
𝑝
9
4. -1 = c – 6
9.
𝑎
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5. John F. Kennedy was the youngest
President to be inaugurated. He was 43
years old. This was 26 years younger
than the oldest president to be
inaugurated – Ronald Reagan. Write
and solve an equation to find out how
old Reagan was when he was
inaugurated.
=9
= −3
10. A shark can swim at an average speed
of 25 miles per hour. At this rate, how far
𝑑
can a shark swim in 2.4 hours? Use r =
𝑡
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Check your UNDERSTANDING
Solve each equation. Check your answer.
Solving equations with Rational Coefficients and Multi-step equation
Find the multiplicative inverse of each
number.
11.
8
5
12. 9
Solve each equation. Check your solution
13. 1.6k = 3.2
14.
3
𝑎
8
=
4
7
12
40
15. -6 = 𝑥
3
16. Dillon deposited of his paycheck into
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the bank. The deposit slip shows how
much he deposited. Write and solve an
equation to find the amount of his
paycheck. Dillon deposited $46.50.
17. 3x + 1 = 7
18. -3y – 5 = 10
19. Syreeta wants to buy some CDs, each
costing $14, and a DVD that costs $23.
She has $65 to spend. Write and solve an
equation to find how many CDs she can
buy.
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Check your UNDERSTANDING
Solve each equation. Check your answer
Solving equations with variables on both sides
Express each equation as another equivalent equation. Justify your answer.
20. 4x + 8 = 2x + 40
21. 9x – 2 = 34 + 3x
Solve each equation. Check your answer
22.
𝑥
3
− 15 = 12 + 𝑥
23. -7 + x = -8 – x
24. 11 + 4x = 7 + 5x
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Check your UNDERSTANDING
Solve each inequality. Check your answer
25. c + 4 < 8
30. 5x > 15
26. 14 + t ≥ 5
31.
27. y – 9 < 11
32. -7y > 28
28. c + 4 ≥ 17
29. An elevator can hold 2,800 pounds or less.
Write and solve an inequality that describes
how much more weight the elevator can hold
if it is currently holding 2,375 pounds.
33.
2
3
4
5
< 𝑦
𝑡
−4
< −11
34. A pool charges $4 each visit, or you can
buy a membership. Write and solve an
inequality to find how many times a person
should use the pool so that a membership is
less expensive than paying each time.
Membership is 3 months for $100.
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