Problem 43 - Algebra and Trigonometry

Question

Solve the inequality. Express the answer using interval notation. $$ \frac{1}{|x+7|}>2 $$

Step-by-Step Solution

Answer

The solution is \((-7.5, -6.5)\).

Step 1: Set up the Inequality

We start with the inequality \( \frac{1}{|x+7|} > 2 \). This means that the reciprocal of the absolute value of \( x+7 \) must be greater than 2.

Step 2: Rewrite the Inequality

To solve \( \frac{1}{|x+7|} > 2 \), we can rewrite it as \( |x+7| < \frac{1}{2} \). This is because for a reciprocal to be greater than a number, the original value must be smaller.

Step 3: Solve the Absolute Value Inequality

The inequality \( |x+7| < \frac{1}{2} \) can be split into two inequalities: \( -\frac{1}{2} < x+7 < \frac{1}{2} \).

Step 4: Solve Both Inequalities

Solve the first inequality: \( -\frac{1}{2} < x+7 \). Subtract 7 from both sides to get \( x > -7.5 \).Solve the second inequality: \( x+7 < \frac{1}{2} \). Subtract 7 from both sides to get \( x < -6.5 \).

Step 5: Write the Solution in Interval Notation

Combine the two inequalities \( -7.5 < x < -6.5 \). In interval notation, this is expressed as \((-7.5, -6.5)\).

Key Concepts

Absolute Value

Understanding absolute values is crucial in solving many types of inequalities. The absolute value of a number, represented as \(|x|\), refers to the distance of that number from zero on the number line, regardless of direction. It is always non-negative.
For example, both 3 and -3 have an absolute value of 3. This is written as \(|-3| = 3\) and \(|3| = 3\). When you solve absolute value inequalities, like \(|x+7| < \frac{1}{2}\), you are finding the range of values for which the expression stays within the specified bounds.
In our example, \(|x+7| < \frac{1}{2}\), this inequality means that the expression \(x+7\) must lie within 0.5 units away from zero. To solve it, we translate it into two separate inequalities:

• \(-\frac{1}{2} < x+7 < \frac{1}{2}\)

Interval Notation

Interval notation is a valuable way to express the solutions of inequalities. It clearly indicates the range of values that satisfy an inequality. In interval notation, a solution set is written in terms of the smallest and largest values in the set, with round or square brackets.
Round brackets \(( )\) indicate that an endpoint is not included in the interval. Square brackets \([ ]\) indicate an endpoint is included. For example, \((-2, 3)\) denotes that all numbers between -2 and 3 (not including -2 or 3) are part of the solution. In contrast, \([-2, 3]\) includes -2 and 3.
In the context of our inequality, the solution \(-7.5 < x < -6.5\) in interval notation is written as \((-7.5, -6.5)\), showing that \(x\) can be any value between -7.5 and -6.5, without including the endpoints.

Algebraic Inequalities

Algebraic inequalities involve expressions containing variables compared using inequality signs like \(<\), \(>\), \(\leq\), or \(\geq\). These inequalities are solved to find the set of possible values for the variables that satisfy the given condition.
The process begins by manipulating the inequality through algebraic operations such as addition, subtraction, multiplication, or division. Care must be taken, especially when multiplying or dividing by negative numbers, as those actions reverse the inequality sign.
For example, in the inequality \(-\frac{1}{2} < x+7 < \frac{1}{2}\), solving each part separately involves isolating the variable by performing operations like subtraction. The end goal is to find a range of values for \(x\), which ultimately lead to a solution expressed neatly in interval notation. This step-by-step approach eliminates any potential confusion and ensures accurate results.