Algebra 1 · Unit 1
Solve linear equations in one variable. Distribute, collect like terms, and use the properties of equality. Then check.
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Steps
Easy example
Solve. 5x + 8 = 33
5x + 8 = 33
Subtract 8 from both sides.
5x = 25
Divide both sides by 5.
x = 5
Check.
5(5) + 8 = 25 + 8 = 33
Answer: x = 5
Medium example
Solve. 3(x − 2) = 2x + 5
3(x − 2) = 2x + 5
Distribute.
3x − 6 = 2x + 5
Subtract 2x from both sides.
x − 6 = 5
Add 6 to both sides.
x = 11
Check.
3(11 − 2) = 3(9) = 27
2(11) + 5 = 22 + 5 = 27
Answer: x = 11
Hard example
Solve. 23x − 4 = 16x + 2
23x − 4 = 16x + 2
The LCD is 6. Multiply every term by 6.
4x − 24 = x + 12
Subtract x from both sides.
3x − 24 = 12
Add 24 to both sides.
3x = 36
Divide both sides by 3.
x = 12
Check.
23(12) − 4 = 8 − 4 = 4
16(12) + 2 = 2 + 2 = 4
Answer: x = 12
Bloblems
A bloblem asks you to find the value of a group such as x + 2, not the value of x. Treat the group as one blob. Combine the blobs, then divide. Do not distribute and solve for x first.
Solve for x + 2. 4(x + 2) + 5(x + 2) = 90
4(x + 2) + 5(x + 2) = 90
Treat (x + 2) as one group.
Combine like terms.
9(x + 2) = 90
Divide both sides by 9.
x + 2 = 10
Check.
4(10) + 5(10) = 40 + 50 = 90
Answer: x + 2 = 10
Two steps. Then check.
5x + 8 = 33
Subtract 8 from both sides.
5x = 25
Divide both sides by 5.
x = 5
Check.
5(5) + 8 = 25 + 8 = 33
Answer: x = 5
7x − 9 = 26
Add 9 to both sides.
7x = 35
Divide both sides by 7.
x = 5
Check.
7(5) − 9 = 35 − 9 = 26
Answer: x = 5
x4 + 3 = 8
Subtract 3 from both sides.
x4 = 5
Multiply both sides by 4.
x = 20
Check.
204 + 3 = 5 + 3 = 8
Answer: x = 20
3x + 11 = 2
Subtract 11 from both sides.
3x = −9
Divide both sides by 3.
x = −3
Check.
3(−3) + 11 = −9 + 11 = 2
Answer: x = −3
8 = 2x − 6
Add 6 to both sides.
14 = 2x
Divide both sides by 2.
x = 7
Check.
2(7) − 6 = 14 − 6 = 8
Answer: x = 7
6x + 5 = 29
Subtract 5 from both sides.
6x = 24
Divide both sides by 6.
x = 4
Check.
6(4) + 5 = 24 + 5 = 29
Answer: x = 4
Variables and numbers on both sides.
4x − 3 = 2x + 11
Subtract 2x from both sides.
2x − 3 = 11
Add 3 to both sides.
2x = 14
Divide both sides by 2.
x = 7
Check.
4(7) − 3 = 28 − 3 = 25
2(7) + 11 = 14 + 11 = 25
Answer: x = 7
5x + 2 = 3x + 14
Subtract 3x from both sides.
2x + 2 = 14
Subtract 2 from both sides.
2x = 12
Divide both sides by 2.
x = 6
Check.
5(6) + 2 = 30 + 2 = 32
3(6) + 14 = 18 + 14 = 32
Answer: x = 6
7x − 4 = 3x + 16
Subtract 3x from both sides.
4x − 4 = 16
Add 4 to both sides.
4x = 20
Divide both sides by 4.
x = 5
Check.
7(5) − 4 = 35 − 4 = 31
3(5) + 16 = 15 + 16 = 31
Answer: x = 5
6x + 1 = 2x + 17
Subtract 2x from both sides.
4x + 1 = 17
Subtract 1 from both sides.
4x = 16
Divide both sides by 4.
x = 4
Check.
6(4) + 1 = 24 + 1 = 25
2(4) + 17 = 8 + 17 = 25
Answer: x = 4
8x − 5 = 5x + 10
Subtract 5x from both sides.
3x − 5 = 10
Add 5 to both sides.
3x = 15
Divide both sides by 3.
x = 5
Check.
8(5) − 5 = 40 − 5 = 35
5(5) + 10 = 25 + 10 = 35
Answer: x = 5
9x + 7 = 4x − 8
Subtract 4x from both sides.
5x + 7 = −8
Subtract 7 from both sides.
5x = −15
Divide both sides by 5.
x = −3
Check.
9(−3) + 7 = −27 + 7 = −20
4(−3) − 8 = −12 − 8 = −20
Answer: x = −3
3x − 11 = x + 5
Subtract x from both sides.
2x − 11 = 5
Add 11 to both sides.
2x = 16
Divide both sides by 2.
x = 8
Check.
3(8) − 11 = 24 − 11 = 13
8 + 5 = 8 + 5 = 13
Answer: x = 8
10x + 3 = 6x − 13
Subtract 6x from both sides.
4x + 3 = −13
Subtract 3 from both sides.
4x = −16
Divide both sides by 4.
x = −4
Check.
10(−4) + 3 = −40 + 3 = −37
6(−4) − 13 = −24 − 13 = −37
Answer: x = −4
2x + 15 = 7x − 5
Subtract 7x from both sides.
−5x + 15 = −5
Subtract 15 from both sides.
−5x = −20
Divide both sides by −5.
x = 4
Check.
2(4) + 15 = 8 + 15 = 23
7(4) − 5 = 28 − 5 = 23
Answer: x = 4
4x − 9 = 8x + 7
Subtract 8x from both sides.
−4x − 9 = 7
Add 9 to both sides.
−4x = 16
Divide both sides by −4.
x = −4
Check.
4(−4) − 9 = −16 − 9 = −25
8(−4) + 7 = −32 + 7 = −25
Answer: x = −4
11x + 6 = 5x − 18
Subtract 5x from both sides.
6x + 6 = −18
Subtract 6 from both sides.
6x = −24
Divide both sides by 6.
x = −4
Check.
11(−4) + 6 = −44 + 6 = −38
5(−4) − 18 = −20 − 18 = −38
Answer: x = −4
Distribute first.
2(n + 5) = −2
Distribute.
2n + 10 = −2
Subtract 10 from both sides.
2n = −12
Divide both sides by 2.
n = −6
Check.
2(−6 + 5) = 2(−1) = −2
Answer: n = −6
3(n + 4) = 15
Distribute.
3n + 12 = 15
Subtract 12 from both sides.
3n = 3
Divide both sides by 3.
n = 1
Check.
3(1 + 4) = 3(5) = 15
Answer: n = 1
4(n − 2) = 20
Distribute.
4n − 8 = 20
Add 8 to both sides.
4n = 28
Divide both sides by 4.
n = 7
Check.
4(7 − 2) = 4(5) = 20
Answer: n = 7
5(n + 3) = 10
Distribute.
5n + 15 = 10
Subtract 15 from both sides.
5n = −5
Divide both sides by 5.
n = −1
Check.
5(−1 + 3) = 5(2) = 10
Answer: n = −1
2(n − 7) = −8
Distribute.
2n − 14 = −8
Add 14 to both sides.
2n = 6
Divide both sides by 2.
n = 3
Check.
2(3 − 7) = 2(−4) = −8
Answer: n = 3
6(n + 1) = 18
Distribute.
6n + 6 = 18
Subtract 6 from both sides.
6n = 12
Divide both sides by 6.
n = 2
Check.
6(2 + 1) = 6(3) = 18
Answer: n = 2
−3(n + 2) = 9
Distribute.
−3n − 6 = 9
Add 6 to both sides.
−3n = 15
Divide both sides by −3.
n = −5
Check.
−3(−5 + 2) = −3(−3) = 9
Answer: n = −5
2(n + 8) = −4
Distribute.
2n + 16 = −4
Subtract 16 from both sides.
2n = −20
Divide both sides by 2.
n = −10
Check.
2(−10 + 8) = 2(−2) = −4
Answer: n = −10
7(n − 1) = 21
Distribute.
7n − 7 = 21
Add 7 to both sides.
7n = 28
Divide both sides by 7.
n = 4
Check.
7(4 − 1) = 7(3) = 21
Answer: n = 4
8(n + 3) = 8
Distribute.
8n + 24 = 8
Subtract 24 from both sides.
8n = −16
Divide both sides by 8.
n = −2
Check.
8(−2 + 3) = 8(1) = 8
Answer: n = −2
5(n − 4) = −15
Distribute.
5n − 20 = −15
Add 20 to both sides.
5n = 5
Divide both sides by 5.
n = 1
Check.
5(1 − 4) = 5(−3) = −15
Answer: n = 1
Distribute, then variables on both sides.
3(x − 2) = 2x + 5
Distribute.
3x − 6 = 2x + 5
Subtract 2x from both sides.
x − 6 = 5
Add 6 to both sides.
x = 11
Check.
3(11 − 2) = 3(9) = 27
2(11) + 5 = 22 + 5 = 27
Answer: x = 11
4(x + 1) = 2x + 14
Distribute.
4x + 4 = 2x + 14
Subtract 2x from both sides.
2x + 4 = 14
Subtract 4 from both sides.
2x = 10
Divide both sides by 2.
x = 5
Check.
4(5 + 1) = 4(6) = 24
2(5) + 14 = 10 + 14 = 24
Answer: x = 5
5(x − 3) = 3x + 1
Distribute.
5x − 15 = 3x + 1
Subtract 3x from both sides.
2x − 15 = 1
Add 15 to both sides.
2x = 16
Divide both sides by 2.
x = 8
Check.
5(8 − 3) = 5(5) = 25
3(8) + 1 = 24 + 1 = 25
Answer: x = 8
2(x + 7) = x + 9
Distribute.
2x + 14 = x + 9
Subtract x from both sides.
x + 14 = 9
Subtract 14 from both sides.
x = −5
Check.
2(−5 + 7) = 2(2) = 4
−5 + 9 = −5 + 9 = 4
Answer: x = −5
6(x − 1) = 4x + 8
Distribute.
6x − 6 = 4x + 8
Subtract 4x from both sides.
2x − 6 = 8
Add 6 to both sides.
2x = 14
Divide both sides by 2.
x = 7
Check.
6(7 − 1) = 6(6) = 36
4(7) + 8 = 28 + 8 = 36
Answer: x = 7
3(x + 4) = x − 2
Distribute.
3x + 12 = x − 2
Subtract x from both sides.
2x + 12 = −2
Subtract 12 from both sides.
2x = −14
Divide both sides by 2.
x = −7
Check.
3(−7 + 4) = 3(−3) = −9
−7 − 2 = −7 − 2 = −9
Answer: x = −7
4(x − 5) = 2x − 6
Distribute.
4x − 20 = 2x − 6
Subtract 2x from both sides.
2x − 20 = −6
Add 20 to both sides.
2x = 14
Divide both sides by 2.
x = 7
Check.
4(7 − 5) = 4(2) = 8
2(7) − 6 = 14 − 6 = 8
Answer: x = 7
5(x + 2) = 3x + 16
Distribute.
5x + 10 = 3x + 16
Subtract 3x from both sides.
2x + 10 = 16
Subtract 10 from both sides.
2x = 6
Divide both sides by 2.
x = 3
Check.
5(3 + 2) = 5(5) = 25
3(3) + 16 = 9 + 16 = 25
Answer: x = 3
2(x − 8) = 5x − 4
Distribute.
2x − 16 = 5x − 4
Subtract 5x from both sides.
−3x − 16 = −4
Add 16 to both sides.
−3x = 12
Divide both sides by −3.
x = −4
Check.
2(−4 − 8) = 2(−12) = −24
5(−4) − 4 = −20 − 4 = −24
Answer: x = −4
7(x + 1) = 4x + 19
Distribute.
7x + 7 = 4x + 19
Subtract 4x from both sides.
3x + 7 = 19
Subtract 7 from both sides.
3x = 12
Divide both sides by 3.
x = 4
Check.
7(4 + 1) = 7(5) = 35
4(4) + 19 = 16 + 19 = 35
Answer: x = 4
3(x − 6) = x + 4
Distribute.
3x − 18 = x + 4
Subtract x from both sides.
2x − 18 = 4
Add 18 to both sides.
2x = 22
Divide both sides by 2.
x = 11
Check.
3(11 − 6) = 3(5) = 15
11 + 4 = 11 + 4 = 15
Answer: x = 11
Distribute and combine like terms.
2(3x + 1) − 4 = 5x + 7
Distribute.
6x + 2 − 4 = 5x + 7
6x − 2 = 5x + 7
Subtract 5x from both sides.
x − 2 = 7
Add 2 to both sides.
x = 9
Check.
2(3(9) + 1) − 4 = 2(28) − 4 = 56 − 4 = 52
5(9) + 7 = 45 + 7 = 52
Answer: x = 9
3(2x + 1) + 5 = 4x + 16
Distribute.
6x + 3 + 5 = 4x + 16
6x + 8 = 4x + 16
Subtract 4x from both sides.
2x + 8 = 16
Subtract 8 from both sides.
2x = 8
Divide both sides by 2.
x = 4
Check.
3(2(4) + 1) + 5 = 3(9) + 5 = 27 + 5 = 32
4(4) + 16 = 16 + 16 = 32
Answer: x = 4
4(x + 3) − 6 = 2x + 8
Distribute.
4x + 12 − 6 = 2x + 8
4x + 6 = 2x + 8
Subtract 2x from both sides.
2x + 6 = 8
Subtract 6 from both sides.
2x = 2
Divide both sides by 2.
x = 1
Check.
4(1 + 3) − 6 = 4(4) − 6 = 16 − 6 = 10
2(1) + 8 = 2 + 8 = 10
Answer: x = 1
2(4x − 1) + 3 = 5x + 10
Distribute.
8x − 2 + 3 = 5x + 10
8x + 1 = 5x + 10
Subtract 5x from both sides.
3x + 1 = 10
Subtract 1 from both sides.
3x = 9
Divide both sides by 3.
x = 3
Check.
2(4(3) − 1) + 3 = 2(11) + 3 = 22 + 3 = 25
5(3) + 10 = 15 + 10 = 25
Answer: x = 3
5(x − 2) + 4 = 3x + 6
Distribute.
5x − 10 + 4 = 3x + 6
5x − 6 = 3x + 6
Subtract 3x from both sides.
2x − 6 = 6
Add 6 to both sides.
2x = 12
Divide both sides by 2.
x = 6
Check.
5(6 − 2) + 4 = 5(4) + 4 = 20 + 4 = 24
3(6) + 6 = 18 + 6 = 24
Answer: x = 6
3(2x − 3) − 1 = 4x + 2
Distribute.
6x − 9 − 1 = 4x + 2
6x − 10 = 4x + 2
Subtract 4x from both sides.
2x − 10 = 2
Add 10 to both sides.
2x = 12
Divide both sides by 2.
x = 6
Check.
3(2(6) − 3) − 1 = 3(9) − 1 = 27 − 1 = 26
4(6) + 2 = 24 + 2 = 26
Answer: x = 6
4(2x + 1) − 5 = 5x + 8
Distribute.
8x + 4 − 5 = 5x + 8
8x − 1 = 5x + 8
Subtract 5x from both sides.
3x − 1 = 8
Add 1 to both sides.
3x = 9
Divide both sides by 3.
x = 3
Check.
4(2(3) + 1) − 5 = 4(7) − 5 = 28 − 5 = 23
5(3) + 8 = 15 + 8 = 23
Answer: x = 3
2(3x − 4) + 5 = 4x − 1
Distribute.
6x − 8 + 5 = 4x − 1
6x − 3 = 4x − 1
Subtract 4x from both sides.
2x − 3 = −1
Add 3 to both sides.
2x = 2
Divide both sides by 2.
x = 1
Check.
2(3(1) − 4) + 5 = 2(−1) + 5 = −2 + 5 = 3
4(1) − 1 = 4 − 1 = 3
Answer: x = 1
6(x + 2) − 10 = 3x + 8
Distribute.
6x + 12 − 10 = 3x + 8
6x + 2 = 3x + 8
Subtract 3x from both sides.
3x + 2 = 8
Subtract 2 from both sides.
3x = 6
Divide both sides by 3.
x = 2
Check.
6(2 + 2) − 10 = 6(4) − 10 = 24 − 10 = 14
3(2) + 8 = 6 + 8 = 14
Answer: x = 2
3(x + 5) − 4 = 2x + 13
Distribute.
3x + 15 − 4 = 2x + 13
3x + 11 = 2x + 13
Subtract 2x from both sides.
x + 11 = 13
Subtract 11 from both sides.
x = 2
Check.
3(2 + 5) − 4 = 3(7) − 4 = 21 − 4 = 17
2(2) + 13 = 4 + 13 = 17
Answer: x = 2
5(2x + 1) − 9 = 6x + 8
Distribute.
10x + 5 − 9 = 6x + 8
10x − 4 = 6x + 8
Subtract 6x from both sides.
4x − 4 = 8
Add 4 to both sides.
4x = 12
Divide both sides by 4.
x = 3
Check.
5(2(3) + 1) − 9 = 5(7) − 9 = 35 − 9 = 26
6(3) + 8 = 18 + 8 = 26
Answer: x = 3
Clear the fractions, then solve.
23x − 4 = 16x + 2
The LCD is 6. Multiply every term by 6.
4x − 24 = x + 12
Subtract x from both sides.
3x − 24 = 12
Add 24 to both sides.
3x = 36
Divide both sides by 3.
x = 12
Check.
23(12) − 4 = 8 − 4 = 4
16(12) + 2 = 2 + 2 = 4
Answer: x = 12
34x + 2 = 12x + 8
The LCD is 4. Multiply every term by 4.
3x + 8 = 2x + 32
Subtract 2x from both sides.
x + 8 = 32
Subtract 8 from both sides.
x = 24
Check.
34(24) + 2 = 18 + 2 = 20
12(24) + 8 = 12 + 8 = 20
Answer: x = 24
12x + 3 = 14x + 7
The LCD is 4. Multiply every term by 4.
2x + 12 = x + 28
Subtract x from both sides.
x + 12 = 28
Subtract 12 from both sides.
x = 16
Check.
12(16) + 3 = 8 + 3 = 11
14(16) + 7 = 4 + 7 = 11
Answer: x = 16
25x − 1 = 15x + 3
The LCD is 5. Multiply every term by 5.
2x − 5 = x + 15
Subtract x from both sides.
x − 5 = 15
Add 5 to both sides.
x = 20
Check.
25(20) − 1 = 8 − 1 = 7
15(20) + 3 = 4 + 3 = 7
Answer: x = 20
56x + 1 = 13x + 4
The LCD is 6. Multiply every term by 6.
5x + 6 = 2x + 24
Subtract 2x from both sides.
3x + 6 = 24
Subtract 6 from both sides.
3x = 18
Divide both sides by 3.
x = 6
Check.
56(6) + 1 = 5 + 1 = 6
13(6) + 4 = 2 + 4 = 6
Answer: x = 6
32x − 5 = 12x + 3
The LCD is 2. Multiply every term by 2.
3x − 10 = x + 6
Subtract x from both sides.
2x − 10 = 6
Add 10 to both sides.
2x = 16
Divide both sides by 2.
x = 8
Check.
32(8) − 5 = 12 − 5 = 7
12(8) + 3 = 4 + 3 = 7
Answer: x = 8
54x − 2 = 12x + 1
The LCD is 4. Multiply every term by 4.
5x − 8 = 2x + 4
Subtract 2x from both sides.
3x − 8 = 4
Add 8 to both sides.
3x = 12
Divide both sides by 3.
x = 4
Check.
54(4) − 2 = 5 − 2 = 3
12(4) + 1 = 2 + 1 = 3
Answer: x = 4
13x + 4 = 56x − 2
The LCD is 6. Multiply every term by 6.
2x + 24 = 5x − 12
Subtract 5x from both sides.
−3x + 24 = −12
Subtract 24 from both sides.
−3x = −36
Divide both sides by −3.
x = 12
Check.
13(12) + 4 = 4 + 4 = 8
56(12) − 2 = 10 − 2 = 8
Answer: x = 12
35x − 2 = 110x + 1
The LCD is 10. Multiply every term by 10.
6x − 20 = x + 10
Subtract x from both sides.
5x − 20 = 10
Add 20 to both sides.
5x = 30
Divide both sides by 5.
x = 6
Check.
35(6) − 2 = 3 − 2 = 1
110(6) + 1 = 0 + 1 = 1
Answer: x = 6
23(x − 6) = 12x + 1
The LCD is 6. Multiply every term by 6.
4(x − 6) = 3x + 6
Distribute.
4x − 24 = 3x + 6
Subtract 3x from both sides.
x − 24 = 6
Add 24 to both sides.
x = 30
Check.
23(30 − 6) = 23(24) = 16
12(30) + 1 = 15 + 1 = 16
Answer: x = 30
34(x + 4) = 12x + 5
The LCD is 4. Multiply every term by 4.
3(x + 4) = 2x + 20
Distribute.
3x + 12 = 2x + 20
Subtract 2x from both sides.
x + 12 = 20
Subtract 12 from both sides.
x = 8
Check.
34(8 + 4) = 34(12) = 9
12(8) + 5 = 4 + 5 = 9
Answer: x = 8
Clear the decimals, then solve.
0.2x + 1.4 = 0.5x − 0.4
Multiply every term by 10.
2x + 14 = 5x − 4
Subtract 5x from both sides.
−3x + 14 = −4
Subtract 14 from both sides.
−3x = −18
Divide both sides by −3.
x = 6
Check.
0.2(6) + 1.4 = 1.2 + 1.4 = 2.6
0.5(6) − 0.4 = 3 − 0.4 = 2.6
Answer: x = 6
0.3x + 1.2 = 0.7x − 0.4
Multiply every term by 10.
3x + 12 = 7x − 4
Subtract 7x from both sides.
−4x + 12 = −4
Subtract 12 from both sides.
−4x = −16
Divide both sides by −4.
x = 4
Check.
0.3(4) + 1.2 = 1.2 + 1.2 = 2.4
0.7(4) − 0.4 = 2.8 − 0.4 = 2.4
Answer: x = 4
0.5x − 1.5 = 0.2x + 0.9
Multiply every term by 10.
5x − 15 = 2x + 9
Subtract 2x from both sides.
3x − 15 = 9
Add 15 to both sides.
3x = 24
Divide both sides by 3.
x = 8
Check.
0.5(8) − 1.5 = 4 − 1.5 = 2.5
0.2(8) + 0.9 = 1.6 + 0.9 = 2.5
Answer: x = 8
0.4x + 0.8 = 0.1x + 2
Multiply every term by 10.
4x + 8 = x + 20
Subtract x from both sides.
3x + 8 = 20
Subtract 8 from both sides.
3x = 12
Divide both sides by 3.
x = 4
Check.
0.4(4) + 0.8 = 1.6 + 0.8 = 2.4
0.1(4) + 2 = 0.4 + 2 = 2.4
Answer: x = 4
1.2x − 0.6 = 0.4x + 1.8
Multiply every term by 10.
12x − 6 = 4x + 18
Subtract 4x from both sides.
8x − 6 = 18
Add 6 to both sides.
8x = 24
Divide both sides by 8.
x = 3
Check.
1.2(3) − 0.6 = 3.6 − 0.6 = 3
0.4(3) + 1.8 = 1.2 + 1.8 = 3
Answer: x = 3
0.25x + 1 = 0.75x − 2
Multiply every term by 100.
25x + 100 = 75x − 200
Subtract 75x from both sides.
−50x + 100 = −200
Subtract 100 from both sides.
−50x = −300
Divide both sides by −50.
x = 6
Check.
0.25(6) + 1 = 1.5 + 1 = 2.5
0.75(6) − 2 = 4.5 − 2 = 2.5
Answer: x = 6
0.6x + 2.4 = 1.2x − 0.6
Multiply every term by 10.
6x + 24 = 12x − 6
Subtract 12x from both sides.
−6x + 24 = −6
Subtract 24 from both sides.
−6x = −30
Divide both sides by −6.
x = 5
Check.
0.6(5) + 2.4 = 3 + 2.4 = 5.4
1.2(5) − 0.6 = 6 − 0.6 = 5.4
Answer: x = 5
0.8x − 1.6 = 0.3x + 0.9
Multiply every term by 10.
8x − 16 = 3x + 9
Subtract 3x from both sides.
5x − 16 = 9
Add 16 to both sides.
5x = 25
Divide both sides by 5.
x = 5
Check.
0.8(5) − 1.6 = 4 − 1.6 = 2.4
0.3(5) + 0.9 = 1.5 + 0.9 = 2.4
Answer: x = 5
0.15x + 0.45 = 0.05x + 1.05
Multiply every term by 100.
15x + 45 = 5x + 105
Subtract 5x from both sides.
10x + 45 = 105
Subtract 45 from both sides.
10x = 60
Divide both sides by 10.
x = 6
Check.
0.15(6) + 0.45 = 0.9 + 0.45 = 1.35
0.05(6) + 1.05 = 0.3 + 1.05 = 1.35
Answer: x = 6
2.5x + 1.5 = 0.5x + 7.5
Multiply every term by 10.
25x + 15 = 5x + 75
Subtract 5x from both sides.
20x + 15 = 75
Subtract 15 from both sides.
20x = 60
Divide both sides by 20.
x = 3
Check.
2.5(3) + 1.5 = 7.5 + 1.5 = 9
0.5(3) + 7.5 = 1.5 + 7.5 = 9
Answer: x = 3
0.4x − 1.2 = −0.2x + 2.4
Multiply every term by 10.
4x − 12 = −2x + 24
Add 2x to both sides.
6x − 12 = 24
Add 12 to both sides.
6x = 36
Divide both sides by 6.
x = 6
Check.
0.4(6) − 1.2 = 2.4 − 1.2 = 1.2
−0.2(6) + 2.4 = −1.2 + 2.4 = 1.2
Answer: x = 6
4(2x − 3) − (x + 5) = 3(x − 1) + 2
Distribute.
8x − 12 − x − 5 = 3x − 3 + 2
7x − 17 = 3x − 1
Subtract 3x from both sides.
4x − 17 = −1
Add 17 to both sides.
4x = 16
Divide both sides by 4.
x = 4
Check.
4(2(4) − 3) − (4 + 5)
4(8 − 3) − 9
4(5) − 9
20 − 9
11
3(4 − 1) + 2
3(3) + 2
9 + 2
11
Answer: x = 4
23(x − 3) − 12(x + 2) = 16(2x − 12) + 1
The LCD is 6. Multiply every term by 6.
4(x − 3) − 3(x + 2) = 1(2x − 12) + 6
Distribute.
4x − 12 − 3x − 6 = 2x − 12 + 6
x − 18 = 2x − 6
Subtract x from both sides.
−18 = x − 6
Add 6 to both sides.
−12 = x
Check.
23(−12 − 3) − 12(−12 + 2)
23(−15) − 12(−10)
−10 − (−5)
−5
16(2(−12) − 12) + 1
16(−24 − 12) + 1
16(−36) + 1
−6 + 1
−5
Answer: x = −12
0.2(5x − 15) − 0.4(x + 5) = 0.1(2x − 8) + 0.6
Multiply every term by 10.
2(5x − 15) − 4(x + 5) = 1(2x − 8) + 6
Distribute.
10x − 30 − 4x − 20 = 2x − 8 + 6
6x − 50 = 2x − 2
Subtract 2x from both sides.
4x − 50 = −2
Add 50 to both sides.
4x = 48
Divide both sides by 4.
x = 12
Check.
0.2(5(12) − 15) − 0.4(12 + 5)
0.2(60 − 15) − 0.4(17)
0.2(45) − 6.8
9 − 6.8
2.2
0.1(2(12) − 8) + 0.6
0.1(24 − 8) + 0.6
0.1(16) + 0.6
1.6 + 0.6
2.2
Answer: x = 12
x + 94 = x + 911
Treat (x + 9) as one group.
The LCD is 44. Multiply every term by 44.
11(x + 9) = 4(x + 9)
Subtract 4(x + 9) from both sides.
7(x + 9) = 0
Divide both sides by 7.
x + 9 = 0
Subtract 9 from both sides.
x = −9
Check.
−9 + 94 = 04 = 0
−9 + 911 = 011 = 0
Answer: x = −9
2x − 83 = 2x − 87
Treat (2x − 8) as one group.
The LCD is 21. Multiply every term by 21.
7(2x − 8) = 3(2x − 8)
Subtract 3(2x − 8) from both sides.
4(2x − 8) = 0
Divide both sides by 4.
2x − 8 = 0
Add 8 to both sides.
2x = 8
Divide both sides by 2.
x = 4
Check.
8 − 83 = 03 = 0
8 − 87 = 07 = 0
Answer: x = 4
3x + 125 = 3x + 128
Treat (3x + 12) as one group.
The LCD is 40. Multiply every term by 40.
8(3x + 12) = 5(3x + 12)
Subtract 5(3x + 12) from both sides.
3(3x + 12) = 0
Divide both sides by 3.
3x + 12 = 0
Subtract 12 from both sides.
3x = −12
Divide both sides by 3.
x = −4
Check.
−12 + 125 = 05 = 0
−12 + 128 = 08 = 0
Answer: x = −4
Solve for the group, not for x. Combine the blobs, then divide.
4(x + 2) + 5(x + 2) = 90
Treat (x + 2) as one group.
Combine like terms.
9(x + 2) = 90
Divide both sides by 9.
x + 2 = 10
Check.
4(10) + 5(10) = 40 + 50 = 90
Answer: x + 2 = 10
3(x + 7) + 5(x + 7) = 56
Treat (x + 7) as one group.
Combine like terms.
8(x + 7) = 56
Divide both sides by 8.
x + 7 = 7
Check.
3(7) + 5(7) = 21 + 35 = 56
Answer: x + 7 = 7
8(n − 3) − 3(n − 3) = 45
Treat (n − 3) as one group.
Combine like terms.
5(n − 3) = 45
Divide both sides by 5.
n − 3 = 9
Check.
8(9) − 3(9) = 72 − 27 = 45
Answer: n − 3 = 9
4(2x + 1) + (2x + 1) = 30
Treat (2x + 1) as one group.
Combine like terms.
5(2x + 1) = 30
Divide both sides by 5.
2x + 1 = 6
Check.
4(6) + 6 = 24 + 6 = 30
Answer: 2x + 1 = 6
7(x + 4) = 2(x + 4) + 35
Treat (x + 4) as one group.
Subtract 2(x + 4) from both sides.
5(x + 4) = 35
Divide both sides by 5.
x + 4 = 7
Check.
7(7) = 49
2(7) + 35 = 14 + 35 = 49
Answer: x + 4 = 7
6(y − 2) + 3(y − 2) + 8 = 44
Treat (y − 2) as one group.
Combine like terms.
9(y − 2) + 8 = 44
Subtract 8 from both sides.
9(y − 2) = 36
Divide both sides by 9.
y − 2 = 4
Check.
6(4) + 3(4) + 8 = 24 + 12 + 8 = 44
Answer: y − 2 = 4
Fractions of a group, blobs on both sides, or find a related expression.
x + 34 + x + 32 = 9
Treat (x + 3) as one group.
The LCD is 4. Multiply every term by 4.
(x + 3) + 2(x + 3) = 36
Combine like terms.
3(x + 3) = 36
Divide both sides by 3.
x + 3 = 12
Check.
124 + 122 = 3 + 6 = 9
Answer: x + 3 = 12
x − 15 + x − 110 = 6
Treat (x − 1) as one group.
The LCD is 10. Multiply every term by 10.
2(x − 1) + (x − 1) = 60
Combine like terms.
3(x − 1) = 60
Divide both sides by 3.
x − 1 = 20
Check.
205 + 2010 = 4 + 2 = 6
Answer: x − 1 = 20
4(x + 6) + 3(x + 6) = 56
Treat (x + 6) as one group.
Combine like terms.
7(x + 6) = 56
Divide both sides by 7.
x + 6 = 8
Now find 2(x + 6).
2(x + 6)
2(8)
16
Check.
4(8) + 3(8) = 32 + 24 = 56
Answer: 2(x + 6) = 16
5(2x − 3) − 4 = 2(2x − 3) + 14
Treat (2x − 3) as one group.
Subtract 2(2x − 3) from both sides.
3(2x − 3) − 4 = 14
Add 4 to both sides.
3(2x − 3) = 18
Divide both sides by 3.
2x − 3 = 6
Check.
5(6) − 4 = 30 − 4 = 26
2(6) + 14 = 12 + 14 = 26
Answer: 2x − 3 = 6
3(x + 9) + 8 = (x + 9) − 6
Treat (x + 9) as one group.
Subtract (x + 9) from both sides.
2(x + 9) + 8 = −6
Subtract 8 from both sides.
2(x + 9) = −14
Divide both sides by 2.
x + 9 = −7
Check.
3(−7) + 8 = −21 + 8 = −13
−7 − 6 = −13
Answer: x + 9 = −7
2(3x − 4) − 5 = 11
Treat (3x − 4) as one group.
Add 5 to both sides.
2(3x − 4) = 16
Divide both sides by 2.
3x − 4 = 8
Now find 5(3x − 4) + 1.
5(3x − 4) + 1
5(8) + 1
40 + 1
41
Check.
2(8) − 5 = 16 − 5 = 11
Answer: 5(3x − 4) + 1 = 41
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