Solving linear inequalities is a lot like solving linear equations, but with a few extra rules to keep in mind. Here’s a step-by-step guide to help you through the process.
- Simplify Both Sides
Start by simplifying both sides of the inequality. This means distributing any multiplication and combining like terms. For example, if you have $3(x + 2) > 2x + 5$, distribute the 3 and combine like terms:
$3x + 6 > 2x + 5$
- Move Variable Terms to One Side
Next, get all the variable terms on one side of the inequality and the constant terms on the other. Subtract $2x$ from both sides in our example:
$3x – 2x + 6 > 5$
This simplifies to:
$x + 6 > 5$
- Isolate the Variable
Now, isolate the variable by performing inverse operations. Subtract 6 from both sides:
$x > 5 – 6$
This simplifies to:
$x > -1$
- Reverse the Inequality Sign When Multiplying or Dividing by a Negative Number
One special rule to remember: if you multiply or divide both sides of the inequality by a negative number, you must reverse the inequality sign. For instance, if you have $-2x < 6$, divide both sides by -2 and reverse the inequality sign:
$x > -3$
- Write the Solution in Interval Notation (Optional)
Sometimes, it’s helpful to express your solution in interval notation. For $x > -1$, the interval notation is:
$(-1, infty)$
- Graph the Solution (Optional)
Graphing the solution on a number line can also be useful. For $x > -1$, draw an open circle at -1 and shade everything to the right.
Example Problem
Let’s solve another example: $4 – 5x leq 9$. Follow the steps:
- Simplify both sides (already simplified).
- Move variable terms to one side:
$-5x leq 9 – 4$
$-5x leq 5$ - Isolate the variable by dividing by -5 and reverse the inequality sign:
$x geq -1$ - Interval notation: $[-1, infty)$
- Graph the solution: closed circle at -1, shade to the right.
Conclusion
By following these steps, you can solve any linear inequality. Remember to always reverse the inequality sign when multiplying or dividing by a negative number. With practice, solving these inequalities will become second nature.