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Chapter 4: Problem 46

Solve the equations and inequalities. $$\frac{x+3}{5}-x=2$$

Short Answer

Expert verified
The solution is \(x = - \frac{7}{4}\).

Step by step solution

01

Clear the fraction

Multiply each term by 5 to get rid of the fraction: \(\frac{x + 3}{5} \times 5 - x \times 5 = 2 \times 5\). This simplifies to \(x + 3 - 5x = 10\).
02

Combine like terms

Combine the \(x\) terms on the left side of the equation: \(x - 5x + 3 = 10\). This simplifies to \(-4x + 3 = 10\).
03

Isolate the variable term

Subtract 3 from both sides of the equation: \(-4x + 3 - 3 = 10 - 3\). This simplifies to \(-4x = 7\).
04

Solve for \(x\)

Divide both sides by -4: \(\frac{-4x}{-4} = \frac{7}{-4}\). This gives \(x = - \frac{7}{4}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

clearing fractions
When solving linear equations, fractions can sometimes make the problem seem complex. To make your work easier, it's best to get rid of (or clear out) the fractions first. This is done by multiplying every term in the equation by the same number. For instance, if we have an equation like: $$\frac{x+3}{5}-x=2$$, we can clear the fraction by multiplying every term by 5. This step will change our equation to:
  • $$\frac{x + 3}{5} \times 5 - x \times 5 = 2 \times 5$$
Simplifying the equation, we get:
  • $$x + 3 - 5x = 10$$
Now, the fractions are gone, and the equation looks simpler!
combining like terms
Once the fractions are cleared, the next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In the example:
  • $$x + 3 - 5x = 10$$
To combine like terms, group the terms with the variable x on one side:
  • $$x - 5x + 3 = 10$$
This simplifies to:
  • $$-4x + 3 = 10$$
By combining like terms, we make the equation simpler and easier to solve.
isolating variables
When solving equations, our goal is often to find the value of the variable. To do this, we need to isolate the variable on one side of the equation. From the previous step, we have:
  • $$-4x + 3 = 10$$
To isolate $$x$$, we need to get rid of the constant term on the left side. We subtract 3 from both sides:
  • $$-4x + 3 - 3 = 10 - 3$$
This simplifies to:
  • $$-4x = 7$$
Now, the variable term $$-4x$$ is isolated.
algebraic operations
The final steps often involve basic algebraic operations to find the value of the variable. From our isolated variable equation:
  • $$-4x = 7$$
We need to solve for $$x$$. Since $$-4$$ is being multiplied by $$x$$, we need to divide both sides by $$-4$$:
  • $$\frac{-4x}{-4} = \frac{7}{-4}$$
This gives us:
  • $$x = -\frac{7}{4}$$
By using algebraic operations, we've solved for $$x$$ and found that $$x = -\frac{7}{4}$$.

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Most popular questions from this chapter

Set up an equation or inequality and solve the problem. Be sure to indicate clearly what quantity your variable represents. Round to the nearest tenth where necessary. If David walks at the rate of \(5 \mathrm{mph}\) and jogs at the rate of \(9 \mathrm{mph}\) and it takes 2 hours to cover a distance of 16 miles, how much time was spent jogging?

Solve the equations and inequalities. $$0.035 x+0.052(x+5,000)=1,130$$

In each of the following exercises, perform the indicated operations. Express your answer as a single fraction reduced to lowest terms. $$\frac{2}{5 x y}+\frac{1}{10 x^{2}}$$

What is a least common denominator? Why is it needed?

Use a calculator to do the computation. Round off to three decimal places where necessary. $$\frac{1}{8} \cdot\left(\frac{2}{5} \div \frac{1}{2}\right)$$
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