91Ó°ÊÓ

Solving inequalities involves finding the values of a variable that make the inequality true. For this specific problem, we deal with a compound inequality, where we have two separate inequalities combined with 'and'. The main steps are:

• Split the compound inequality into two simpler inequalities.
• Focus on solving each inequality separately by isolating the variable of interest.
• Combine the obtained solutions.

For example, in the given inequality \(-2 \leq \frac{x+2}{-5} \leq 6\), we split it into two parts: \(-2 \leq \frac{x+2}{-5}\) and \(\frac{x+2}{-5} \leq 6\). For each part, isolate \(x\) by performing operations like multiplication, division, addition, or subtraction.
One crucial rule is to reverse the inequality sign when multiplying or dividing by a negative number. For instance, multiplying \(-2 \leq \frac{x+2}{-5}\) by -5 results in \(10 \geq x+2\) which simplifies to \(x \leq 8\). Similarly, solving the second part \(\frac{x+2}{-5} \leq 6\) involves multiplying by -5 to get \(x+2 \geq -30\), resulting in \(x \geq -32\). Combining both solutions, we obtain \(-32 \leq x \leq 8\).

Graphing inequalities on a number line visually shows the range of solution sets. This is especially useful for compound inequalities like \(-32 \leq x \leq 8\).

To graph such an inequality:

• Draw a number line and mark the significant points, here -32 and 8.
• Use closed circles on these points since the inequalities include \(\text{≤}\).
• Shade the region between these two points.

The closed circles indicate that -32 and 8 are part of the solution.
For instance, the inequality \(-32 \leq x \leq 8\) means any value between -32 and 8, inclusive, satisfies the inequality. Graphing helps visually confirm all possible solutions, making it easier to understand and verify our solution set.

A number line is a straight line with numbers placed at equal intervals or segments along its length. It's a simple yet powerful tool to visually represent numbers, inequalities, and ranges of values.

When graphing solutions of inequalities:

• Draw a horizontal line and mark equal intervals.
• Identify and mark points of significance (intersection points).
• Use symbols like closed circles (for \( \leq, \geq \)) or open circles (for \( <, > \)).
• Shade the section of the line that represents all possible solutions.

If \(a \leq x \leq b\), place closed circles at \(a\) and \(b\) and shade the part of the line between them. For example, to represent \(-32 \leq x \leq 8\), place closed circles at -32 and 8 and shade the region in between. This effectively communicates the entire set of solutions in an easy-to-understand format.