Problem 27 - Algebra and Trigonometry

Question

Solve the inequality. Express the answer using interval notation. $$ |x+5| \geq 2 $$

Step-by-Step Solution

Answer

The solution is \((-\infty, -7] \cup [-3, \infty)\).

Step 1: Understand the Absolute Value Property

An absolute value inequality like \(|x+5| \geq 2\) means the expression inside the absolute value, \(x + 5\), is either greater than or equal to 2 or less than or equal to -2. This gives us two distinct inequalities to solve: \(x+5 \geq 2\) and \(x+5 \leq -2\).

Step 2: Solve the First Inequality

Solve \(x+5 \geq 2\) by subtracting 5 from both sides:\[ x+5 - 5 \geq 2 - 5 \]\[ x \geq -3 \].

Step 3: Solve the Second Inequality

Solve \(x+5 \leq -2\) by subtracting 5 from both sides:\[ x+5 - 5 \leq -2 - 5 \]\[ x \leq -7 \].

Step 4: Interpret the Solutions

The solutions to the inequalities are \(x \geq -3\) and \(x \leq -7\). However, as these came from an absolute value inequality, the solution set includes all \(x\) values that satisfy either inequality.

Step 5: Express the Solution in Interval Notation

The two solution sets from the inequalities are \([-\infty, -7]\) and \([-3, \infty]\). Since the absolute value inequality is "greater than or equal to," the complete solution is the union of these two intervals, expressed as: \[ (-\infty, -7] \cup [-3, \infty) \].

Key Concepts

Inequality Solutions

When dealing with an absolute value inequality, you often need to break it down into two separate inequalities. This is because the absolute value measures the distance from zero, meaning the number inside could be either a positive or a negative value. For example, if you have \(|x+5| \geq 2\), it implies two possibilities: either \(x+5\geq 2\) or \(x+5\leq -2\). Each of these needs to be solved individually.
- In the first case, solve \({x+5 \geq 2}\) by isolating the variable. Subtract 5 from both sides to find that \(x \geq -3\).
- In the second case, you solve \(x+5 \leq -2\) in a similar fashion. Subtract 5 from both sides to get \(x \leq -7\).

These solutions indicate that any value of \(x\) that satisfies either inequality is part of the solution set for the original absolute value inequality.

Interval Notation

Interval notation is a shorthand used to describe sets of numbers along a continuous line. It tells you the smallest and largest values in a set, as well as if the endpoints are included or not.
- Square brackets \[a, b\] indicate that both endpoints are included in the set, representing values between and including \(a\) and \(b\).
- Parentheses \(a, b\) imply that neither endpoint is included, signifying numbers strictly between \(a\) and \(b\).

In our solution to the absolute value inequality \(|x+5| \geq 2\), we ended up with two intervals: \([-\infty, -7]\) and \([-3, \infty)\). The union of these two intervals, \((-fty, -7] \cup [-3, \fty)\), indicates that any number in either interval is a valid solution. The interval from \(-\infty\) to \(-7\) includes \(-7\), while the interval from \(-3\) to \(fty\) inclusively begins at \(-3\).

Solving Inequalities

Solving inequalities involves finding a range of values for a variable that makes an inequality true. Unlike solving equal equations, inequalities often result in a span of possible numbers rather than a single number.

To solve the given inequality \(|x+5| \geq 2\), you need to consider both directions of the inequality. Essentially, you're determining not just where one value stands in relation to others, but all possible values within a certain range.
- **Direct Approach**: Start by isolating the variable on one side. If needed, perform operations such as addition, subtraction, multiplication, or division across the inequality to simplify it. Remember that multiplying or dividing an inequality by a negative number flips the inequality symbol.
- **Analysis**: Review whether each section involves a 'greater than,' 'less than,' 'greater than or equal to,' or 'less than or equal to' scenario, modifying how you express the solution.

Ultimately, once all possibilities are handled, you integrate these ranges into one complete solution, sometimes using interval notation to clearly communicate the solution's scope.