91Ó°ÊÓ

Understanding how to solve inequalities is fundamental in algebra. Similar to equations, inequalities also contain an unknown variable, but instead of an equals sign, they use inequality symbols such as 'greater than' (>), 'less than' (<), 'greater than or equal to' (≥), or 'less than or equal to' (≤). The process of solving an inequality is to isolate the variable on one side of the inequality sign to find its range of values, rather than a single solution.

Consider the compound inequality from our exercise involving two separate inequalities: \(2x > -10\) and \(9x < 18\). The solution involves dividing both sides of each inequality by the coefficients of \(x\) (2 and 9, respectively), yielding \(x > -5\) and \(x < 2\). It's important to note that when dividing or multiplying by a negative number in an inequality, the inequality sign must be flipped. However, in this case, since we're dividing by positive numbers, the direction of the inequalities remains unchanged.

Graphing inequalities allows us to visually represent the solution set on a number line or a coordinate plane, depending on whether the inequality is one-dimensional or two-dimensional. A number line is sufficient for most single-variable inequalities.

In the textbook exercise, after solving the inequalities, we need to graph them. On a number line, we often use open circles to indicate that a number is not included in the solution set ('greater than' or 'less than') and closed circles when it is included ('greater than or equal to' or 'less than or equal to').

For the inequality \(x > -5\), an open circle would be placed at -5, and a shaded line would extend rightward. For \(x < 2\), another open circle goes at 2, with a shaded line extending leftward. If both conditions must be met simultaneously, we only highlight the segment common to both, that is, the overlap.

The term 'inequality intersections' refers to the range of values that satisfy more than one inequality simultaneously. When inequalities are connected by the word 'and', such as in compound inequalities, their intersection is the overlapping range where both inequalities are true.

In our exercise, the intersection is found by looking for values that are greater than -5 and at the same time less than 2. Visually on the graph, it's the segment of the number line between the open circles at -5 and 2. This intersection represents all values of \(x\) that make the compound inequality true. Remember, 'and' means both conditions must be true for the same value of \(x\), leading to a smaller range of valid values compared to when inequalities are combined with 'or'.