Problem 17 - Algebra Form and Function

Question

Each of the inequalities can be solved by performing two operations on both sides. State the operations in order of use and solve the inequality. $$ \frac{5}{2} r-r<6 $$

Step-by-Step Solution

Answer

Question: Solve the inequality \(\frac{5}{2} r - r < 6\) and express the solution in terms of the variable r. Answer: \(r < 4\)

Step 1: Rewrite and simplify the inequality

Combine the like terms on the left side of the inequality: $$ \frac{5}{2} r - r = \frac{5}{2} r - \frac{2}{2} r = \frac{3}{2} r < 6 $$ Now the inequality is in a simpler form: $$ \frac{3}{2} r < 6 $$

Step 2: Solve the inequality

We need to isolate r on the left side of the inequality. To do this, first, multiply both sides by the reciprocal of the coefficient of r, which in this case, is \(\frac{2}{3}\): $$ \frac{2}{3} \cdot \frac{3}{2}r < 6 \cdot \frac{2}{3} $$ This simplifies to: $$ r < 4 $$ The solution to the inequality is \(r < 4\).

Key Concepts

Simplifying Expressions

Simplifying expressions is like tidying up your room — putting things in order so you can easily find what you’re looking for. Imagine the expression \(\frac{5}{2} r - r\). It looks a bit messy because you have the same variable \(r\) but with different coefficients. To simplify, we need to perform operations to make it more concise and readable.

Start by addressing the coefficients of the \(r\). You have \(\frac{5}{2}r\) and need to subtract \(\frac{2}{2}r\) (which is equivalent to \(r\)). When you subtract the fractions, make sure your terms have a common denominator. Here they already do, so it's just a matter of subtraction:

\[\frac{5}{2} r - \frac{2}{2} r = \frac{3}{2} r\]

This step leaves us with a simplified expression \(\frac{3}{2} r\). Clear expressions help solve equations easily, just like a clean room helps you find what you need quickly.

Combining Like Terms

Imagine you are sorting through fruit baskets and want to group all the apples together. In algebra, 'combining like terms' is similar. It involves grouping and simplifying expressions by adding or subtracting terms with the same variable.

Let's look again at the expression \(\frac{5}{2} r - r\). Both terms have 'r', so they are like terms. When combining them, focus solely on the coefficients — the numerical part in front of the variable. Subtract \(1r\) (expressed as \(\frac{2}{2} r\) to match the denominator) from \(\frac{5}{2} r\), simplifying it to \(\frac{3}{2} r\).

Here's a quick guide to combining terms:

• Identify like terms: they have the exact variable to the same power.
• Adjust fractions to have common denominators if needed.
• Add or subtract the coefficients.

Combining like terms reduces complexity and provides a clearer path to solving equations and inequalities.

Reciprocal

If you've ever needed to reverse a process, you already understand the basic concept of the reciprocal. In math, the reciprocal of a number is a helpful tool used to simplify divisions and solve equations. Specifically, the reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\). When a number is multiplied by its reciprocal, the result is 1.

In solving the inequality \(\frac{3}{2} r < 6\), we need \(r\) alone on one side to understand how small or large it can be. To do this, we multiply both sides by the reciprocal of \(\frac{3}{2}\), which is \(\frac{2}{3}\). This operation cancels out the fraction on the left side, isolating \(r\):

\[\frac{2}{3} \cdot \frac{3}{2} r = r\;\;\text{on the left, and}\;\; 6 \times \frac{2}{3} = 4\;\;\text{on the right.}\]

So \(r < 4\). Using reciprocals is crucial when the goal is to isolate a variable, making it essential for solving inequalities and equations efficiently.