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Inequalities are mathematical expressions involving the symbols >, <, ≥, or ≤. Solving inequalities is akin to solving equations, but with one key difference—when multiplying or dividing both sides by a negative number, the inequality sign must be reversed.
An inequality like 6 > -x essentially means that 6 is greater than the negative of some value, x. To solve this, multiply both sides by -1. Doing so gives -6 < x because the inequality flips when multiplying by a negative number.
Next, consider the compound inequality given in the problem:

• -6 < x: This part of the inequality tells us x is greater than -6.
• -x ≥ 0: This part tells us -x is non-negative, translating to x ≤ 0.

Combining these two, we get a single compound inequality: -6 < x ≤ 0. This compound inequality tells us that x is between -6 and 0, including 0 but not -6.

Interval notation is a shorthand way of writing subsets of the real number line. It uses brackets and parentheses to describe the set of all numbers between given bounds.
For the inequality -6 < x ≤ 0:

• The open parenthesis '(' at -6 indicates that -6 is not included in the interval.
• The closed bracket ']' at 0 indicates that 0 is included in the interval.

The interval notation for the inequality -6 < x ≤ 0 is written as (-6, 0]. This means x is any number greater than -6 and up to and including 0. It is a concise and useful way to represent solutions to inequalities.

Graphing compound inequalities involves marking the solutions on a number line. This visual representation makes it easier to understand the range of values that satisfy the inequality.
For the given inequality -6 < x ≤ 0:

• First, draw a number line and mark the points -6 and 0.
• Place an open circle at -6 to indicate it is not included in the solution set.
• Place a closed circle at 0 to indicate it is included.
• Shade the region between -6 and 0 to show all the numbers that satisfy the inequality.

This shaded region represents all the values that x can take, ensuring you visually grasp the set of solutions.
Graphing is an incredibly useful tool that helps make abstract numerical concepts more concrete and easier to understand.