91Ó°ÊÓ

Inequality solving is a lot like solving equations, but with a twist. Instead of finding one answer, you find a range of values.
The first step is to rewrite the inequality in a standard form. For the inequality \[\frac{r}{r+2} < 2r\], you'd want to get rid of any fractions. This means multiplying both sides by a common denominator.
Next, simplify the resulting expression. Combine like terms and isolate the variable on one side.
Be aware of one key rule: when you multiply or divide by a negative number, you must flip the inequality sign.

• If \[a > b\], then \[-a < -b\]
• Similarly, if \[a < b\], then \[-a > -b\]

Remember to always check both solutions when dealing with inequalities, especially with variables in the denominator.
This helps to ensure that values that make denominators zero are excluded from the solution set.

Finding a common denominator is crucial when working with rational inequalities or fractions.
It simplifies the expression so you can easily isolate and solve for the variable.
In the given inequality \[\frac{r}{r+2} < 2r\], the common denominator is \[r+2\].
Multiply every term by this common denominator to eliminate the fraction: \[\frac{r}{r+2} \times (r+2) < 2r \times (r+2)\]
Simplifying yields \[r < 2r(r+2)\].
This transformed inequality is easier to work with.
Always remember to simplify as much as possible after finding a common denominator.
This makes the next steps in solving the inequality straightforward and clear.

Graphing solution sets of inequalities helps visualize the range of values that satisfy the inequality.
After solving the inequality, you get a range of values for \[r\].
For instance, if the inequality \[\frac{r}{r+2} < 2r\] transforms and simplifies to \[r(r+2) > 1\], solve it to find the critical points.

• Make a number line and plot points where the inequality changes.
• Test intervals between these points to know where the inequality holds true.

Shade the regions on the number line where the inequality is satisfied.
If the inequality is strict ( < or > ), use open circles to indicate that endpoint values are not included.
If the inequality is inclusive ( ≤ or ≥ ), use closed circles.
These visual cues clarify which values make the inequality true, helping you understand and verify your solution.