Solutions to Inequalities

 

1. Isolate the variable:

• Move all terms containing the variable to one side of the inequality and constants to the other side.
• For example, in the inequality $2�+3<7$, subtract 3 from both sides to isolate the term with $�$: $2�<4$.

2. Perform operations:

• If there are any operations on the variable (addition, subtraction, multiplication, or division), perform them to simplify the inequality.
• Continuing with the example, divide both sides by 2: $�<2$.

3. Simplify:

• Simplify the inequality as much as possible. In our example, $�<2$ is already simplified.

4. Determine the solution:

• Identify the values of the variable that satisfy the inequality.
• For our example, any value of $�$ less than 2 will satisfy the inequality. The solution can be expressed using interval notation as $(-\mathrm{\infty },2)$ or in set notation as $\{�\mid �<2\}$.

Special cases:

• If you multiply or divide both sides of the inequality by a negative number, the direction of the inequality symbol will change. For example, if you have $-2�>6$, dividing both sides by $-2$ gives $�<-3$.