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Solving inequalities involves finding the range of values for a variable that makes an inequality true. Here, let's dissect the inequality step-by-step to understand how it's solved. First, isolate the variable by moving all other terms to the other side. In the example \(-x - 3 \leq 7\), this means adding 3 to both sides to remove the \(-3\) on the left, simplifying the inequality to \(-x \leq 10\). Once you have isolated the term with \(-x\), the next step is crucial. Multiply or divide each side of the inequality by \(-1\) to solve for \(x\). Importantly, whenever you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. So, \(-x \leq 10\) becomes \(x \geq -10\). This indicates that any number \(x\) equal to or greater than \(-10\) solves the inequality. Key points:

• Isolate the variable on one side.
• Reverse the inequality sign when multiplying or dividing by a negative number.
• Consider all steps carefully to ensure no mistakes are made when transforming the inequality.

Graphing inequalities on a number line helps visualize the range of solutions. It makes understanding inequalities more intuitive. In this step, you illustrate the solutions Start by plotting a point on the number line for the boundary of the inequality. For the solution \(x \geq -10\), place a filled-in circle or a line underneath at \(-10\), denoting that \(-10\) is part of the solution set. This is because \(x\) can be equal to \(-10\) due to the 'greater than or equal to' symbol.Next, draw an arrow from this point extending to the right along the number line. This arrow shows that all numbers greater than \(-10\) are also included in the solution set. To graph an inequality:

• Plot the boundary point with a filled circle if it includes \(\leq\) or \(\geq\).
• Use an arrow to indicate the direction and range of the solution.
• Make sure the graph matches the inequality's meaning. A correct graph clearly displays the inequality's solutions at a glance.

Using a number line is a simple way to visualize and solve mathematical problems, including inequalities. It provides a clear display of where a number lies relative to others. When using a number line for inequalities, the position of numbers is crucial. For instance, the number line allows easy identification of whether numbers meet the condition \(x \geq -10\). By marking \(-10\) and showing an extension of all numbers greater, your graph translates the inequality into a visual format. This helps confirm you're interpreting the inequality correctly and ensuring that all possible solutions are included and apparent.Key points for number line usage:

• Mark critical points clearly, showing which are inclusive or exclusive.
• Use arrows effectively to denote possible solutions.
• Ensure the overall representation is accurate and reflects the inequality's scope.

The number line, when used effectively, is powerful in math for checking and graphing solutions to inequalities.