91Ó°ÊÓ

Solving inequalities is similar to solving equations, but with a few extra rules that apply due to the inequality sign. The aim is to find all possible values of a variable that make the inequality true. In our example \[3x + 7 \geq 13\], we need to isolate the variable and then solve for it. Key things to remember when solving inequalities:

- You can perform the same algebraic operations on both sides of an inequality, just like with equations.
- If you multiply or divide both sides by a negative number, you must flip the inequality sign. For instance, if you have \[-2x < 4\], dividing by \[-2\] will result in \[x > -2\].

To solve an inequality, you will often need to isolate the variable on one side. This means you need to get the variable by itself. In our given inequality \[3x + 7 \geq 13\], we start by isolating the term with the variable. Here are the steps to isolate the variable:

1. Eliminate any constants on the same side as the variable.
2. Perform the same operation on both sides of the inequality to maintain the balance.

In the provided solution:

1. Subtract 7 from both sides: \[3x + 7 - 7 \geq 13 - 7\]
2. Simplify to get: \[3x \geq 6\]

This operation zeroes out the constant term, leaving \[3x\] on one side.

After isolating the variable, you're just a few steps away from solving the inequality. You need to use basic algebraic operations to solve for the variable. In our context, we had \[3x \geq 6\]. Here's how we solve it:

- Divide both sides by 3 to solve for x:
\[\frac{3x}{3} \geq \frac{6}{3}\]

- Simplify the inequality:

\[x \geq 2\]

Algebraic operations include addition, subtraction, multiplication, and division. When handling inequalities, remember these quick tips:

- Maintain the inequality by performing the same operation on both sides.
- Flip the inequality sign when multiplying or dividing by a negative number.

Understanding these operations will make solving inequalities much smoother!