Problem 3 - Algebra 2 and Trigonometry

Question

In \(3-14,\) solve and check each inequality. $$ \frac{a}{4}>\frac{a}{2}+6 $$

Step-by-Step Solution

Answer

The solution to the inequality is \(a < -24\).

Step 1: Eliminate Fractions by Finding a Common Denominator

To make the equation simpler, start by eliminating the fractions. Multiply every term by the least common denominator (LCD) of 4 and 2, which is 4. This gives: \[ 4 \times \left( \frac{a}{4} \right) > 4 \times \left( \frac{a}{2} + 6 \right) \] which simplifies to \[ a > 2a + 24 \].

Step 2: Isolate the Variable on One Side

To isolate the variable, subtract \(2a\) from both sides of the inequality:\[ a - 2a > 2a - 2a + 24 \] which simplifies to \[ -a > 24 \].

Step 3: Solve for the Variable

To solve for \(a\), divide both sides by -1. Remember, dividing or multiplying both sides of an inequality by a negative number reverses the inequality sign, so \[ a < -24 \].

Step 4: Verify the Solution

Check the solution by testing a value less than -24, such as \(a = -25\), in the original inequality \[ \frac{a}{4} > \frac{a}{2} + 6 \].Substituting \(a = -25\) gives \[ \frac{-25}{4} > \frac{-25}{2} + 6 \].This calculates to \[ -6.25 > -12.5 + 6 \] which simplifies to \[ -6.25 > -6.5 \],confirming the inequality is true.

Key Concepts

Common Denominator

When solving inequalities involving fractions, finding a common denominator can simplify the process considerably. Imagine trying to solve an inequality with terms like \(\frac{a}{4}\) and \(\frac{a}{2}\). It can be tricky to work with these different denominators, especially if you're trying to perform operations across terms.

By multiplying each term by the least common denominator (LCD), you can eliminate the fractions entirely. For the fractions \(\frac{a}{4}\) and \(\frac{a}{2}\), the LCD is 4. When you multiply everything by 4, it clears the fractions and makes it more straightforward. You get to work with whole numbers, turning it into an easier inequality to handle.

In our exercise, multiplying both sides by 4 changed the inequality \(\frac{a}{4} > \frac{a}{2} + 6\) into \(a > 2a + 24\), which is much simpler to deal with.

Variable Isolation

Isolating the variable means getting the variable by itself on one side of the inequality. This makes it much easier to identify the solution. Once fractions are dealt with, you can focus on rebuilding the inequality to showcase the variable on its own.

Here’s what we did: you saw \(a > 2a + 24\), and needed to subtract \(2a\) from both sides. Think of it like taking numbers from each side of a balanced scale to keep it even.

After performing this action, the inequality changed to \(-a > 24\). Now, the variable is somewhat isolated, but with a negative sign, which requires one more step to simplify.

Reversing Inequality Sign

Reversing the inequality sign is a crucial step that happens when multiplying or dividing both sides of an inequality by a negative number. This rule might seem unusual at first, but it's an essential part of working with inequalities.

In this particular inequality \(-a > 24\), to solve for \(a\), you need to divide both sides by \(-1\). Dividing by a negative doesn't just affect the signs of the numbers; it also flips the direction of the inequality. Thus, \(-a > 24\) turns into \(a < -24\) after division.

This flipping action is like reversing the balance in a scale after you've adjusted it with a negative factor, ensuring the inequality still holds true.

Inequality Verification

Once you have a potential solution from solving the inequality, verifying it confirms that the solution set makes the inequality true. Verification is the process of choosing a number from the solution set and plugging it back into the original inequality.

For the inequality \(\frac{a}{4} > \frac{a}{2} + 6\), after solving, we found \(a < -24\). To verify, you can pick any number less than \(-24\) (say \(a = -25\)), and substitute it back into the inequality.

When \(a = -25\), the expressions become \(-6.25\) and \(-6.5\). Checking whether \(-6.25 > -6.5\) is true (and it is!), confirms that the solution \(a < -24\) is correct. Remember, verification is like double-checking your work to ensure every step is logically sound.