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Solving inequalities is a process similar to solving equations, but with a crucial difference regarding the signs. When you solve an inequality, you're finding all the possible values that make the inequality true, not just a single value. Inequalities can tell us if one quantity is bigger, smaller, or the same as another. To solve inequalities like \(-x + 2 \leq 5\), you should first manipulate the inequality just like you would with an equation:

• Start by getting terms involving the variable on one side. Here we subtract 2 from both sides: \(-x \leq 3\).
• Next, isolate the variable. When you multiply or divide by a negative number, reverse the inequality sign. So multiplying both sides by -1 gives us \(x \geq -3\).

It's essential to remember that inequalities change direction when they are multiplied or divided by a negative number, which is often a stumbling block for many students. This reversal is what maintains the integrity and truth of the inequality.

Graphing inequalities allows us to visually understand the range of solutions that satisfy the inequality. It's a powerful tool to quickly see all the solutions at once. When graphing inequalities like \(x \geq -3\):

• Start with the point where the inequality is equal — in this case, \(x = -3\).
• Determine whether to use an open or closed dot. Here, because we have "greater than or equal to" (\(\geq\)), we use a closed dot, indicating that \(-3\) is included in the solution set.
• Draw a line or arrow starting at the point and extending in the direction of the inequality. Since \(x\) needs to be greater than \(-3\), the line goes to the right.

By following these steps, the graph becomes a visual confirmation of the inequality solution, making the concepts clearer and more intuitive.

Number lines are a simple yet effective way to represent solutions to inequalities. They help us quickly see the set of numbers that fit a particular inequality rule. For \(x \geq -3\):

• First, locate the number \(-3\) on the number line. This point will be our reference.
• Since \(\geq\) includes equality, place a filled, or closed, dot on \(-3\). This differs from a "greater than" inequality, which would use an open dot.
• Then, draw a line or arrow that extends to the right from \(-3\), showing all values greater than \(-3\) are solutions.

A number line can clearly display the set of solutions and make it easier to understand how inequalities are solved and represented. This practical tool enhances comprehension and can be a quick way for students to verify their work.