Linear inequalities are similar to linear equations but involve an inequality symbol instead of an equal sign. For example, instead of solving an equation like \(2x - 6 = 0\), you might encounter an inequality such as \(2x - 6 \leq 0\). The goal is to find the values of \(x\) that satisfy the inequality.

To solve a linear inequality, you can use the same methods as you would for a linear equation. For instance, starting with \(2x - 6 \leq 0\), you can add 6 to both sides to isolate the term with \(x\), resulting in \(2x \leq 6\). Dividing both sides by 2 gives you \(x \leq 3\). This indicates that \(x\) can take any value less than or equal to 3.

However, when dealing with inequalities, special attention is needed when multiplying or dividing by negative numbers. For example, if you have \(-2x - 6 \leq 0\) and you isolate \(x\) by dividing both sides by \(-2\), you must flip the inequality sign. This means that instead of \(x \leq -3\), you will have \(x \geq -3\). Thus, the solution indicates that \(x\) can be any value greater than or equal to \(-3\).

When graphing the solutions to linear inequalities, the representation differs from that of linear equations. For \(x \leq 3\), you would place a closed circle at 3 on the number line and shade to the left, indicating all values less than or equal to 3. In contrast, for \(x \geq -3\), you would place a closed circle at \(-3\) and shade to the right, indicating all values greater than or equal to \(-3\).

In interval notation, the solution \(x \leq 3\) is expressed as \((-∞, 3]\), where the square bracket indicates that 3 is included in the solution. Similarly, \(x \geq -3\) is expressed as \([-3, ∞)\), with the square bracket indicating that \(-3\) is also included. Understanding these concepts is crucial for effectively solving and graphing linear inequalities.