Problem 80 - Algebra and Trigonometry

Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Solve: \(\frac{1}{3}(x-6)+4 x>0\)

Step-by-Step Solution

Answer

\( x > \frac{3}{13} \text{,or approximately x} > 0.462.

Step 1: Distribute the fraction

Distribute \(\frac{1}{3}\) through \((x - 6)\). This gives \(\frac{1}{3}x - 2\).

Step 2: Combine like terms

Combine \(\frac{1}{3}x\) and \(+4x\). This simplifies to \( \frac{1}{3}x + 4x - 2 > 0 \).

Step 3: Simplify the equation

Convert \(\frac{1}{3}x\) to decimals by finding a common denominator. \( \frac{1}{3}x\text{ is the same as }x/3 \). So, it can be rewritten as: \( \frac{1}{3}x = 0.3333x \text{ or } 0.3333x \). Combine with \(4x\) to get: \( \0.3333x + 4x \).

Step 4: Solve for x

Combine terms and solve for x: \(4.3333x > 2 \). Divide both sides by 4.3333: \(\frac{2}{4.3333}= x \). This simplifies to \(\frac{2}{4.33} \text{is approximately equal to } x \)

Key Concepts

solving linear inequalities

Solving linear inequalities is a fundamental aspect of algebra. It involves finding the values of a variable that satisfy the given inequality. Here, we are tasked with solving \(\frac{1}{3}(x-6)+4x>0\).

To solve this, follow these steps:

• First, we distribute any fractions or coefficients through parentheses.


• Next, combine all like terms to simplify the expression.


• Finally, solve for the variable by isolating it on one side of the inequality.

Notice that solving an inequality is quite similar to solving an equation, but we must remember the crucial rule: if we multiply or divide both sides by a negative number, we must reverse the inequality sign.

combining like terms

Combining like terms allows us to simplify expressions more effectively. Like terms are terms that contain the same variable raised to the same power.

In the inequality \(\frac{1}{3}(x-6)+4x>0\), we distribute \(\frac{1}{3}\) through each term to get \(\frac{1}{3}x - 2\).

Now we have: \(\frac{1}{3}x - 2 + 4x > 0\).
Combine the like terms \(\frac{1}{3}x\) and \(4x\).
Converting \(\frac{1}{3}x\) to decimal form, we get 0.3333x, so we add 0.3333x and 4x:

\(\frac{1}{3}x + 4x \rightarrow 0.3333x + 4x = 4.3333x\). Now, our inequality looks like this: 4.3333x - 2 > 0. By combining like terms, we effectively reduce the complexity of the expression, making it easier to solve.

simplifying expressions

Simplifying expressions involves making them as concise as possible without changing their value. This often involves combining like terms and performing arithmetic operations.

Let's consider the expression we derived: 4.3333x - 2 > 0.

Now, we need to isolate x.

• Add 2 to both sides to get 4.3333x > 2.


• Then, divide both sides by 4.3333 to isolate x.

This gives us:
\ x > \frac{2}{4.3333} \
Approximating, we get \ x > \frac{2}{4.33} = 0.461\

Hence, the solution to the inequality is \( x > 0.461 \). Simplifying expressions is crucial for making calculations more manageable, especially when dealing with inequalities.