Question

Solve each inequality. Express your answer using set notation or interval notation. Graph the solution set. \((4 x+2)^{-1}<0\)

Short answer

The solution is \( x < -\frac{1}{2} \) or in interval notation \( (-\infty, -\frac{1}{2}) \).

Step-by-step solution

Apply the inequality condition

Given the inequality \( (4x+2)^{-1}<0 \), we need to determine the values of \(x\) that satisfy this inequality. Recall that the reciprocal function \( (4x+2)^{-1} \) is negative when \( 4x+2 \) is negative.

Solve the inequality

Set up the inequality \( 4x+2 < 0 \). To solve for \( x \), isolate \( x \) by subtracting 2 from both sides to get \( 4x < -2 \). Then, divide both sides by 4: \[ x < -\frac{1}{2} \].

Express the solution in interval notation

The solution to the inequality \( 4x+2 < 0 \) is all \( x \) less than \( -\frac{1}{2} \). In interval notation, this is expressed as \( (-\infty, -\frac{1}{2}) \).

Graph the solution set

To graph the solution set, draw a number line and shade all the values to the left of \( -\frac{1}{2} \). Place an open circle at \( -\frac{1}{2} \) to indicate that this point is not included in the solution.

Key concepts

Set Notation

Set notation is a way to describe a collection of elements or numbers that satisfy a specific condition. In the context of inequalities, we use set notation to express the solution set of an inequality. Here's how it works: if a variable, say **x**, can take on any value that satisfies the inequality, we place **x** within a pair of curly braces and describe the condition it must satisfy. For example, the inequality **4x + 2 < 0** results in the solution set **{x | x < -1/2}**. This reads as 'the set of all x such that x is less than -1/2'.
Set notation is useful because it gives a precise and compact way to describe the solutions of an inequality.

Interval Notation

Interval notation provides another way to represent sets of numbers, especially those that lie on the number line. It uses intervals to describe the range of values that satisfy an inequality. The endpoints of intervals are indicated using parentheses **()** for non-inclusive bounds and brackets **[]** for inclusive bounds.
For instance, if the solution to an inequality is that **x** must be less than **-1/2**, this is written in interval notation as **(-∞, -1/2)**. The **-∞** symbol means that the interval extends infinitely in the negative direction, and the parenthesis around **-1/2** indicates that **-1/2** is not included in the solution set.

Graphing Inequalities

Graphing inequalities on a number line visually represents the range of values that are solutions to the inequality. To graph the solution set of an inequality like **4x + 2 < 0**:
- First, solve the inequality to find the boundary point (e.g., **x < -1/2**).
- Draw a number line, and mark the boundary point with an open circle if the point is not included (as it is in **x < -1/2**). If the point is included, use a closed dot.
- Shade the region of the number line where the inequality holds. For example, for **x < -1/2**, shade all numbers to the left of **-1/2**. This gives a clear picture of the solution set.

Reciprocal Function

The reciprocal function is denoted by **f(x) = 1/x**. For the given inequality **(4x+2)^{-1} < 0**, this function is in the form of the reciprocal. The key thing to remember is that the reciprocal function is negative when its argument, **4x + 2**, is negative.
Thus, to solve **(4x + 2)^{-1} < 0**, you can set up the inequality **4x + 2 < 0** and solve for **x**. It's essential to solve for the argument first to determine where the reciprocal function will be negative. In this case, solving **4x + 2 < 0** leads to **x < -1/2**. The reciprocal function has given us the necessary condition to solve the original inequality.