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Compound inequalities are a powerful tool in algebra. They allow us to combine two inequalities into one statement. This can often simplify the problem-solving process by giving you a more holistic view of the possible values for your variable. In the context of the original exercise, the absolute value inequality \(|5x + 9| \leq 16\) translates to a compound inequality: \(-16 \leq 5x + 9 \leq 16\). Here, the compound inequality is made up of two separate statements joined by the word "and." Our goal is to satisfy both conditions simultaneously.

• First, we have \(-16 \leq 5x + 9\).
• Second, \(5x + 9 \leq 16\).

Both parts of the inequality have to be true at the same time for any value of \(x\). Understanding the logic behind compound inequalities is crucial for solving problems involving absolute values.

When it comes to solving inequalities, it's essential to follow similar steps as solving equations, with a few key differences. In inequalities, you are often working with a range of values rather than a specific number. Here, for our compound inequality \(-16 \leq 5x + 9 \leq 16\), we handle each part of the compound inequality individually.Let's break it down:

• For \(-16 \leq 5x + 9\), subtract \(9\) from both sides to isolate the term with \(x\), giving \(-25 \leq 5x\).
• Next, divide by \(5\) to solve for \(x\), resulting in \(-5 \leq x\).
• For the other part, \(5x + 9 \leq 16\), similarly subtract \(9\) from both sides to get \(5x \leq 7\).
• Dividing by \(5\), we find \(x \leq \frac{7}{5}\).

Once both inequalities are simplified, the solution is the range of \(x\) values that simultaneously satisfy both inequalities, leading us to \(-5 \leq x \leq \frac{7}{5}\). Remember, when multiplying or dividing an inequality by a negative number, you must reverse the inequality sign. However, in this problem, we focus on positive operations, so no sign reversal is necessary.

Understanding algebraic expressions is essential in solving absolute value inequalities like the one in the exercise. Here, the expression \(5x + 9\) plays a central role and needs careful handling.An algebraic expression, simply put, is a combination of numbers, variables, and operational signs. In our case:

• Number: \(9\)
• Variable term: \(5x\)
• Operations involve addition and multiplication.

When solving the inequality \(|5x + 9| \leq 16\), it is crucial to treat the expression inside the absolute value brackets as a whole. This lets you split the inequality into two separate conditions, each tied to the properties of absolute value. Once you manage the algebraic expression, you can break it into its compound inequality parts. Solving such inequalities becomes a matter of performing algebraic manipulations to isolate \(x\). Keeping a keen eye on changes to the inequality and maintaining balance on both sides of the expression are pivotal skills in effectively managing algebraic expressions.