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Interval notation is a streamlined way to represent the range of solutions of an inequality. It's used widely in algebra to describe all the numbers in an interval along the number line. In interval notation, the solution to an inequality is given in the form of a pair of values that defines the starting point and the endpoint of the interval. For instance, if you have the solution set

• \(x \geq 6\),
• this means all numbers starting from 6 to infinity meet the condition.

This is represented as

• \([6, \infty)\), where "\([\)" indicates that 6 is included.

On the other hand, if an interval excludes a number, you would use "\(()\)." These notations help in clearly conveying if endpoints are part of the solution set or not.
It's essential to always follow these specific signs, as they determine if numbers on the boundary are included or beyond your interval.

Compound inequalities involve more than one inequality statement. They can either be "and" inequalities or "or" inequalities, determining how solutions need to be combined. For an "and" compound inequality, as seen in part (a) of our exercise, both inequalities must hold true simultaneously. Hence,

• \(3x - 2 \geq 4\) (which simplifies to \(x \geq 2\))
• and \(x + 6 \geq 12\) (simplifying to \(x \geq 6\)).
• The overlap, or most restrictive solution, is the solution for both, being \(x \geq 6\).

In "or" inequalities shown in part (b), a solution can be valid if it meets at least one inequality.

• Either \(x \geq 2\)
• or \(x \geq 6\) will suffice.

Because \(x \geq 2\) already covers the whole region of \(x \geq 6\) as well, the solution is therefore \(x \geq 2\), which captures both parts efficiently.

Graphing inequalities can be a powerful visual tool that helps solidify understanding. They provide a quick way to see which part of the number line is part of the solution.
To graph an inequality like \(x \geq 6\), begin by drawing a number line and placing a closed circle over the number 6.
This closed circle indicates that 6 is included in the solution set. Then, shade the entire region to the right of 6 to represent all numbers greater than or equal to 6.

• The shaded area forms your graphical solution.

For the compound inequality \(x \geq 2\), you'll start by placing a closed circle on the number 2 and shading everything to the right of it.
This shading clearly displays any number greater than or equal to 2 fulfills the inequality, encompassing all solutions from the key initial problems we explored.
These graphs are not only proof of the solution but also a handy check to ensure all possibilities are considered in compound inequalities.