Problem 47 - Beginning and Intermediate Algebra

Question

Explain why the solution to \(|2 x+1| \geq-3\) is \((-\infty, \infty)\).

Step-by-Step Solution

Answer

The given inequality is \(|2x + 1| \geq -3\). Since the absolute value of any expression is always non-negative, it's always greater than or equal to 0, and thus always greater than or equal to -3. Therefore, the inequality holds true for all real numbers x, and the solution set is \((-\infty, \infty)\).

Step 1: Understand the properties of absolute values.

Absolute value is a function that measures the distance of a number from zero on the number line. It is defined as the following: \( |x| = \left\{ \begin{array}{ll} x & \mbox{if } x \geq 0 \\ -x & \mbox{if } x < 0 \end{array} \right. \) For any real number x, the absolute value of x is always non-negative. Thus, \(|x| \geq 0\).

Step 2: Original inequality.

We are given the inequality: \( |2x + 1| \geq -3 \)

Step 3: Apply the property of absolute values.

Since the absolute value is always greater than or equal to 0, we can write the following for the given inequality: \( |2x + 1| \geq 0 \) Now, remember that this inequality is true for all real numbers x because an absolute value can never be negative.

Step 4: Compare the two inequalities.

Comparing the two inequalities, we can see that they are equivalent since they both share the same property of absolute values always being greater than or equal to 0. Original inequality: \( |2x + 1| \geq -3 \) Adjusted inequality: \( |2x + 1| \geq 0 \) Since the inequality is true for all real numbers x, the solution set is: \( (-\infty, \infty) \)