Solving Two-Step Equations and Inequalities
Objectives:
…to solve two-step equations involving integers, decimals, and fractions
...to solve two-step inequalities involving integers, decimals, and fractions
...to check possible solutions for two-step equations and inequalities
Assessment Anchor:
8.D.2.1 – Select and/or use a strategy to simplify an expression, solve an
equation or inequality and/or check the solution for accuracy.
NOTES
***EQUATIONS ARE LIKE SEE-SAWS, AND MUST REMAIN BALANCED!!
To solve a two step equation:
1. Locate the variable in the equation
2. Use the inverse (opposite) operation on both sides of the equation
a) FIRST use addition or subtraction to isolate the variable part
b) SECOND use multiplication or division to isolate the variable
3. Show your answer
EXAMPLES
1)
Add 7 on both sides
Divide by 2 on both sides
2x – 7 = -15
-3x – 12 = 18
2x – 7 = -15
+7 +7
2x = -8
2
2
x = -4
-3x – 12 = 18
-3x =
x=
Solving Two-Step Equations and Inequalities
2)
Subtract 10 on both sides
Divide by 4 on both sides
3)
Subtract 19 on both sides
Multiply by 7 on both sides
-4x + 10 = -34
5x + 13 = -33
-4x + 10 = -34
– 10 – 10
-4x = -44
-4
-4
x = 11
5x + 13 = -33
x + 19 = 4
7
x + 40 = -11
2
x + 19 = 4
7
– 19 – 19
7 • x = -15 • 7
7
x = -105
9 = x – 12
-3
4)
Add 12 on both sides
Multiply by 3 on both sides
9 = x – 12
-3
+ 12
+ 12
-3 • 21 = x • -3
-3
-63 = x
5x =
x=
x + 40 = -11
2
x=
2
x=
-24 = x – 9
-5
-24 = x – 9
-5
=x
-5
=x
MORE EXAMPLES
5)
-3x + 11 = -34
6)
-9 = 12 + 5y
Solving Two-Step Equations and Inequalities
7)
9)
x
8
+ 14 = 3
-4x – 18 = -16
8)
10)
y
4
– 20 = -12
45 =
x
8
+ 51
EVEN MORE EXAMPLES – Careful Here!!
11)
22 – 5k = -18
13)
-7 = -21 –
x
4
12)
-9 = 30 – 2y
14)
23 –
x
5
= 32
Solving Two-Step Equations and Inequalities
15)
-32 = 18 – 8k
16)
-11 = -9 –
x
3
STILL MORE EXAMPLES – Decimals and Fractions!!
17)
2x + 4.5 = -2.9
18)
x
3
19)
0.4y + 5.6 = 2.4
20)
1
x
3
+ 21 = 10
21)
3
k
4
22)
2
y
9
+ 21 = -11
– 17 = -11
– 1.93 = -4.1
Solving Two-Step Equations and Inequalities
MORE NOTES
To solve a two step inequality:
1. Follow the same procedure for solving equations.
2. ***IF YOU MULTIPLY OR DIVIDE BY A NEGATIVE NUMBER
ON BOTH SIDES at the very end, YOU MUST SWITCH THE
INEQUALITY SYMBOL.
EXAMPLES
1)
Add 11 on both sides
Divide by 3 on both sides
graph of solution
3x + -11 ≥ -20
+ 11 + 11
3x ≥ -9
3
3
x ≥ -3
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***Symbol stayed the same because you divided by a positive 3 on both sides.
2)
Add 14 on both sides
Divide by -2 on both sides
graph of solution
-2x – 14 < -24
+ 14 + 14
-2x < -10
-2
-2
x>5
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***Symbol switched because you divided by a negative 2 on both sides.
3)
Add 3 on both sides
Divide by 4 on both sides
-19 ≤ -3 + 4x
+3 +3
-16 ≤ 4x
4
4
-4 ≤ x
graph of solution
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***Symbol stayed the same because you divided by a positive 4 on both sides.
Solving Two-Step Equations and Inequalities
MORE EXAMPLES
4)
-3x + 31 < 25
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5)
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-1y + 9 > 7
6)
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-18 – 4x ≥ 4
8)
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– 20 < -23
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30 ≤
9)
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y
2
7)
5
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43 ≥ 8 – 5y
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x
3
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+ 33
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