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When graphing inequalities, it is important to represent the solution set visually on a number line. This helps us understand which numbers satisfy the inequality and which do not. To graph the inequality \(x < -3\), begin by locating the number -3 on the number line. Since the inequality symbol \(<\) indicates that values less than -3 are included, place an open circle at -3. An open circle means that -3 itself is not part of the solution. Draw a line or shade the region to the left of -3. This indicates all numbers less than -3 satisfy the inequality, continuing towards \(-\infty\). Graphing is a useful method to double-check solutions and offers a visual representation of inequalities.

Interval notation is a way of describing the set of solutions to an inequality. It is concise and can be easily understood. For the inequality \(x < -3\), the correct interval notation is \(( -\infty, -3)\). Let’s break down what each part means.

• \((-\infty\)) means that the solution starts from negative infinity, showing that there is no lower bound.
• \(,-3)\) indicates that -3 is the upper limit but is not included in the solution set, as shown by the round parenthesis.

Round brackets \(( )\) in interval notation show that an endpoint is not included, while square brackets \([ ]\) would indicate inclusion. For inequalities with infinity, always use round brackets, as infinity itself is not a reachable number.

Understanding the properties of inequalities helps manipulate and solve them correctly. Here are some critical properties:

• If we add or subtract the same number from both sides of an inequality, the inequality’s direction stays the same. For example, if \(a < b\), then \(a+c < b+c\) for any real number \(c\).
• Multiplying or dividing both sides of an inequality by a positive number leaves the direction unchanged. For example, \(2x < -6\) becomes \(x < -3\) when divided by the positive 2.
• However, multiplying or dividing by a negative number reverses the inequality's direction. If we had \(-2x > -6\), dividing by -2 would change it to \(x < 3\).

It’s essential to carefully apply these properties to solve inequalities accurately. Mistakes like not changing the inequality symbol when dividing by a negative number can lead to incorrect solutions.