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Chapter 6: Problem 77

Solve each equation. Use set notation to express solution sets for equations with no solution or equations that are true for all real numbers. \(7+2(3 x-5)=8-3(2 x+1)\)

Short Answer

Expert verified
The solution to the equation is: \(x = 2/3\)

Step by step solution

01

Apply the Distributive Property

Let's start by applying the distributive property of multiplication over addition/subtraction. On the left side of the equation, \(2 \times (3x - 5)\) becomes \(6x - 10\). Similarly on the right side of the equation, \(-3 \times (2x + 1)\) becomes \(-6x - 3\). So, the equation now transforms into: \(7 + 6x - 10 = 8 - 6x -3\)
02

Simplify both sides

Now, let's simplify both sides of the equation by combining like terms. On the left side, combine \(7\) and \(-10\) to get \(-3\), and on the right side, combine \(8\) and \(-3\), which results in \(5\). So, the equation simplifies into: \(6x - 3 = 5 - 6x\).
03

Arrange terms with \(x\)

Now let's arrange the equation by moving the terms with \(x\) on one side and the constant on the other side. Add \(6x\) on both sides, the equation becomes: \(12x - 3 = 5\)
04

Isolate the variable

To solve for \(x\), we have to isolate it. Let's do this by adding \(3\) on both sides, and the equation becomes: \(12x = 8\)
05

Solve for \(x\)

Lastly, to solve for \(x\), divide by \(12\) on both sides. So, \(x = 8 / 12\), which simplifies into \(x = 2 / 3\)

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Most popular questions from this chapter

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