Problem 57 - Algebra for College Students

Question

Solve each inequality and express the solution set using interval notation. \(-3(3 x+2)-2(4 x+1) \geq 0\)

Step-by-Step Solution

Answer

\((-\infty, -\frac{8}{17}]\)

Step 1: Distribute the constants

First, distribute the constants \(-3\) and \(-2\) across the terms in the parentheses. \(-3(3x+2)\) becomes \(-9x - 6\), and \(-2(4x+1)\) becomes \(-8x - 2\). This gives us the inequality \(-9x - 6 - 8x - 2 \geq 0\).

Step 2: Combine like terms

Combine the x terms and the constant terms on the left side of the inequality. The terms \(-9x\) and \(-8x\) combine to \(-17x\). The constants \(-6\) and \(-2\) combine to \(-8\). This simplifies the inequality to \(-17x - 8 \geq 0\).

Step 3: Isolate the x term

Add 8 to both sides of the inequality to isolate the \(-17x\) term. This gives \(-17x \geq 8\).

Step 4: Solve for x

To find x, divide both sides of the inequality by \(-17\). Remember to reverse the inequality sign when dividing by a negative number. Thus, we get \(-17x / -17 \leq 8 / -17\), which simplifies to \ x \leq -\frac{8}{17} \

Step 5: Express the solution in interval notation

Convert the inequality solution to interval notation. Since \ x \leq -\frac{8}{17} \, the interval notation is \(-\infty, -\frac{8}{17}]\). This interval includes all numbers less than or equal to \ -\frac{8}{17}\.

Key Concepts

Solving Inequalities

Inequalities are like equations, but instead of an equals sign, they use symbols such as \(<\), \(>\), \(\leq\), or \(\geq\) to show the relationship between expressions. When solving inequalities, the goal is to isolate the variable, much like solving equations. However, there is a key difference concerning inequalities: when you multiply or divide both sides by a negative number, the inequality sign must be flipped.

Consider the inequality \(-3(3x+2)-2(4x+1) \geq 0\). The solution involves:

• Distributing the constants across the terms in parentheses to eliminate them.
• Combining like terms to simplify the inequality.
• Isolating the variable to one side of the inequality to solve for it.

Each step must be carefully executed to maintain the balance of the inequality, especially when dealing with negative signs during multiplication or division.

Interval Notation

Interval notation is a way to describe the set of solutions to an inequality. It is compact and clear, showing the range of values that satisfy the inequality. In interval notation, a parenthesis \(()\) indicates that an endpoint is not included in the set, while a square bracket \([]\) indicates that it is included. For instance, \(-\infty, a\) represents all numbers less than \(a\), whereas \(-\infty, a]\) means all numbers less than or equal to \(a\).

In our exercise, the solution \(x \leq -\frac{8}{17}\) translates to \(-\infty, -\frac{8}{17}]\) in interval notation. This notation means any number less than or equal to \(-\frac{8}{17}\) is included in the solution set. The use of \(-\infty\) indicates that there is no lower bound to the values included, meaning they extend indefinitely in the negative direction.

Combining Like Terms

Combining like terms is a process regularly used to simplify mathematical expressions, especially within inequalities. Like terms are terms that contain the same variable raised to the same power. For example, \(-9x\) and \(-8x\) can be combined because they both have the variable \(x\).

To combine them, you simply add their coefficients: \(-9 + (-8) = -17\). The expression then becomes \(-17x\). Similarly, with constant terms like \(-6\) and \(-2\), you add them together to get \(-8\). This helps in condensing expressions and making it easier to isolate the variable.

• Combining like terms simplifies the inequality, reducing complexity.
• It enhances clarity and sets the stage for the next steps in solving the inequality.
• Failure to properly combine like terms can lead to errors in solving the inequality.

Engaging carefully with like terms ensures each step aligns towards achieving a clear resolution of the inequality.