91Ó°ÊÓ

Inequality solutions refer to the range of values that satisfy a given inequality expression. In our exercise, we dealt with a compound inequality that involves two separate inequalities connected by the word 'or'.

The first inequality is \( x \text{ } \text{≥} \text{ } 5 \). This means any value of \( x \) that is greater than or equal to 5 is a solution. To solve it, we simply need values where \( x \) is 5 or more.

The second inequality is \( -x \text{ } \text{≥} \text{ } 4 \). This needs an extra step. We multiply both sides by -1. Remember, multiplying or dividing by a negative number reverses the inequality sign. So, \( -x \text{ } \text{≥} \text{ } 4 \) becomes \( x \text{ } \text{≤} \text{ } -4 \). Here, any value where \( x \) is less than or equal to -4 works.

The compound nature of inequalities means we combine these two sets of solutions. One that includes all values greater than or equal to 5 and one that includes all values less than or equal to -4.

Interval notation is a way to describe sets of numbers or solutions to inequalities. It uses brackets and parentheses to indicate where intervals start and end.

For instance, consider the inequality \( x \text{ } \text{≥} \text{ } 5 \). In interval notation, we represent this as \( [5, \text{ } \text{∞}) \). The square bracket [ means the number 5 is included, and the parenthesis ) means the interval goes on indefinitely towards positive infinity.

Now look at the inequality \( x \text{ } \text{≤} \text{ } -4 \). We write this in interval notation as \( (-\text{∞}, \text{ } -4] \). Here, the parenthesis ( indicates the interval extends infinitely in the negative direction, and the square bracket ] shows that -4 is included in the interval.

For the whole compound inequality, we combine the two intervals:\( (-\text{∞}, \text{ } -4] \text{ } \text{∪} \text{ } [5, \text{ } \text{∞}) \). The union symbol \( ∪ \) means that the combined intervals include the values where \( x \) is either less than or equal to -4 or greater than or equal to 5.

Graphing inequalities involves plotting the solution range on a number line. This visual representation helps us understand the solution set for the inequality.

Start by plotting critical points mentioned in the solutions. In our example: -4 and 5. To show they are included in the solution, we use closed dots on these points.

For the compound inequality \( x \text{ } \text{≥} \text{ } 5 \), shade the number line to the right of 5. This shows all values starting from 5 going rightward to positive infinity.

For \( x \text{ } \text{≤} \text{ } -4 \), shade the number line to the left of -4. This indicates all values starting from -4 moving leftward to negative infinity.

The two shaded regions combined show the entire solution set to the compound inequality, which can also be represented by the interval notation \( (-\text{∞}, \text{ } -4] ∪ [5, \text{ } ∞) \).

This graphing technique helps students visually verify if their solution matches the mathematical intervals.