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Solving inequalities involves a few key principles that are similar to solving equations. The main objective is to isolate the variable on one side of the inequality sign. For the given inequality, we start with \[ 5x > -45 \]To solve for \( x \), divide both sides by 5. This operation must be applied equally to both sides to maintain the inequality:- Divide the left side: \( \frac{5x}{5} = x \)- Divide the right side: \( \frac{-45}{5} = -9 \)Thus, the inequality simplifies to:\[ x > -9 \]Important to note is that when dividing or multiplying both sides of an inequality by a negative number, the inequality sign flips direction. In this exercise, we did not need to do that because we divided by a positive number.

Graphing inequalities is an integral step for visually interpreting the solution of an inequality. It helps in understanding the range of possible values for the variable involved. In this context, the solution \( x > -9 \) indicates that \( x \) can be any real number greater than -9.

When graphing this on a number line:

• Start by marking the critical point, which is -9 in this case.
• Since -9 is not included in the solution (indicated by the \( > \) rather than \( \geq \)), represent it as an open circle.
• Shade or draw an arrow extending to the right of -9 indicating all numbers greater than -9.

By following these steps, we graphically represent that the solution encompasses all values to the right of -9 on the number line.

Number line representation of inequalities gives a clear visual depiction of the solution set. This can especially help in understanding the range and boundaries of solutions within a given context. For our inequality solution, the number line effectively shows all possible values of \( x \) that satisfy \( x > -9 \).

Here are the steps to represent it on a number line:

• Identify the critical value from the inequality, here it is -9.
• Place an open circle on -9 to signify that it is not included in the solution.
• Draw an arrow pointing to the right from -9 to show every number greater than -9 is included in the solution set.

The ease of seeing which values make the inequality true makes number line representation a valuable tool in understanding inequalities.