solve 7(x + 4) - 2/3 (x - 6) < 2 (x - 3(x + 5)
7(x+4) - 2/3 (x-6) < 2(x-3(x + 5))
7x+28 - 2/3 x + 4 < 2x - 6x - 30
19/3 x + 32 < -4x - 30
31/3 x < -62
x < -6
To solve the inequality 7(x + 4) - 2/3 (x - 6) < 2 (x - 3(x + 5)), we can follow these steps:
Step 1: Distribute and simplify both sides of the inequality:
7x + 28 - (2/3)(x) + 4 - (2/3)(-6) < 2x - 6(x) - 3(5x + 15)
Simplifying further:
7x + 28 - (2/3)x + 4 + 4/3 < 2x - 6x - 15x - 45
Step 2: Combine like terms:
7x - (2/3)x + 32/3 < -19x - 45
Step 3: Get rid of fractions by multiplying through by the least common multiple (LCM) of the denominators, which in this case is 3. This step is optional since we can also solve the inequality with fractions:
3(7x - (2/3)x) + 3(32/3) < 3(-19x - 45)
Simplifying further:
21x - 2x + 32 < -57x - 135
Step 4: Combine like terms:
19x + 32 < -57x - 135
Step 5: Move all terms with x to one side and constants to the other side:
19x + 57x < -135 - 32
76x < -167
Step 6: Divide both sides by 76 (the coefficient of x) to solve for x:
x < -167/76
So the solution to the inequality 7(x + 4) - 2/3 (x - 6) < 2 (x - 3(x + 5)) is x < -167/76.
To solve the inequality 7(x + 4) - 2/3(x - 6) < 2(x - 3(x + 5)), we need to simplify the expression and isolate the variable x.
Step 1: Distribute and simplify
7(x + 4) - 2/3(x - 6) < 2(x - 3(x + 5))
7x + 28 - (2/3)x + 4 < 2x - 6(x + 5)
Step 2: Combine like terms
7x + 28 - (2/3)x + 4 < 2x - 6x - 30
To combine the terms on the left side of the inequality, we need to find a common denominator for 2/3 and -30.
Step 2.1: Find a common denominator
The common denominator for 2/3 and -30 is 3. Multiply 2/3 by 3/3 to get 6/9.
7x + 28 - (2/3)x + 4 < 2x - 6x - 30
7x + 28 - (6/9)x + 4 < 2x - 6x - 30
Step 3: Simplify further
7x + 28 - (6/9)x + 4 < 2x - 6x - 30
Combine the x terms: 7x - (6/9)x + 2x - 6x
Combine the constant terms: 28 + 4 + 30
1x + 32 - (6/9)x < -4x - 26
Step 4: Combine the x terms and the constant terms
1x - (6/9)x + 4x < -4x - 58
To combine the x terms, we need to find a common denominator for 1 and -6/9.
Step 4.1: Find a common denominator
The common denominator for 1 and -6/9 is 9. Multiply 1 by 9/9 to get 9/9.
9/9x - (6/9)x + 4x < -4x - 58
Step 5: Simplify further
9/9x - (6/9)x + 4x < -4x - 58
Combine the x terms: 9/9x - (6/9)x + 4x - (-4x)
Combine the constant terms: -58
(9/9 - 6/9 + 4)x < - 58
Simplify further: (3/9)x < -58
Step 6: Simplify the inequality
To simplify (3/9)x < -58, we can divide both sides of the inequality by 3/9.
(3/9)x / (3/9) < -58 / (3/9)
Simplify further: x < -58 / (3/9)
Step 7: Divide and simplify
To divide by a fraction, we can multiply by its reciprocal.
x < -58 * (9/3)
Simplify further: x < -174
So, the solution to the inequality 7(x + 4) - 2/3(x - 6) < 2(x - 3(x + 5)) is x < -174.