Problem 35 - Algebra for College Students

Question

Solve each equation and inequality. \(|5 x+9| \leq 16\)

Step-by-Step Solution

Answer

The solution is \[-5 \leq x \leq \frac{7}{5}\].

Step 1: Understand the Absolute Value Inequality

An absolute value inequality such as \[|5x + 9| \leq 16\] means that the expression inside the absolute value can be between \[-16\] and \[16\] inclusive. This expresses that the distance from zero is less than or equal to \[16.\]

Step 2: Set Up the Compound Inequality

Rewrite the absolute value inequality as a compound inequality without the absolute value.\[-16 \leq 5x + 9 \leq 16\]This shows two separate inequalities combined into one. The expression \[5x + 9\] must be greater than or equal to \[-16\] and less than or equal to \[16.\]

Step 3: Solve the First Inequality

Let's solve the first inequality:\[-16 \leq 5x + 9\]Subtract \[9\] from both sides:\[-16 - 9 \leq 5x\] which simplifies to:\[-25 \leq 5x\]Now, divide by \[5\]:\[-5 \leq x\]

Step 4: Solve the Second Inequality

Now, solve the second part of the compound inequality:\[5x + 9 \leq 16\]Subtract \[9\] from both sides:\[5x \leq 16 - 9\]This simplifies to:\[5x \leq 7\]Divide by \[5\]:\[x \leq \frac{7}{5}\]

Step 5: Combine the Solutions

Combine the solutions from Step 3 and Step 4 to get the final solution for the inequality:\[-5 \leq x \leq \frac{7}{5}\]This means \[x\] can be any value between \[-5\] and \[\frac{7}{5}\], inclusive.

Key Concepts

Compound Inequalities

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.

Solving Inequalities

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.

Algebraic Expressions

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.