Problem 64 - Precalculus Student Solutions Manual 5th

Question

Solve each equation or inequality. $$|3 x+2| \leq 0$$

Step-by-Step Solution

Answer

x = -\frac{2}{3}

Step 1: Understanding Absolute Value Inequality

An absolute value inequality such as \(|3x + 2| \leq 0 \) means we need to find the values of \(x\) for which the expression inside the absolute value is either equal to or less than zero. However, note that absolute values are always either zero or positive.

Step 2: Analyze the Inequality

Since absolute values are always non-negative, the inequality \( |3x + 2| \leq 0 \) means that \( |3x + 2|\) must be equal to 0. Because it can never be less than 0.

Step 3: Set the Expression Inside the Absolute Value to Zero

Set the inside of the absolute value equal to zero: \( 3x + 2 = 0 \).

Step 4: Solve for \(x\)

Solve the equation: \( 3x + 2 = 0 \. ewline 3x = -2 \ewline x = - \frac{2}{3} \).

Step 5: Verify the Solution

Verify that the solution satisfies the original inequality: \( |3(-\frac{2}{3}) + 2| = | -2 + 2 | = 0 \leq 0 \). This is true.

Key Concepts

Solving Inequalities

Solving inequalities involves finding the range of values for the variable that makes the inequality true. An inequality expresses a relationship where one side does not equal the other. They can be '<' (less than), '<=' (less than or equal to), '>' (greater than), or '>=' (greater than or equal to).
To solve inequalities:

• Isolate the variable on one side.
• Perform the same operations on both sides, similar to solving equations but with extra care around multiplication or division by negative numbers.
• Graphical solutions can visually represent the inequality on a number line.

An important rule: when you multiply or divide both sides of an inequality by a negative number, the inequality sign flips. This is crucial because not adhering to this rule can lead to incorrect solutions. Understanding and applying these rules help solve inequalities accurately.

Absolute Value

The absolute value of a number represents the distance from zero on a number line, regardless of direction. It is always non-negative and denoted by two vertical lines, for example, \(|x|\).
For absolute value inequalities, we explore two main scenarios:

• Inequalities with '<=' or '>=':

  For example, |A| <= B, implies -B <= A <= B.



• Inequalities with '<' or '>':

  |A| < B, implies -B < A < B.

Let's apply this to \(|3x + 2| <= 0\). Since the absolute value cannot be negative, the only way this is true is if the expression inside equals zero. Thus, 3x + 2 must be zero.

Linear Equations

Linear equations are equations of the first order. They involve variables raised only to the power of one and graph as straight lines. A standard form is \(Ax + B = C\), where A, B, and C are constants.
To solve linear equations, isolate the variable:

• Combine like terms on each side.
• Use addition or subtraction to move terms involving the variable to one side.
• Use multiplication or division to solve for the variable.

In our exercise, we see \(3x + 2 = 0\). To isolate x:

• Subtract 2 from both sides: \(3x = -2\).
• Divide both sides by 3: \(x = -\frac{2}{3}\).

Verification involves plugging \(x = -\frac{2}{3}\) back into the original inequality to ensure it holds true.