Real numbers

Real Numbers

Real numbers include all rational and irrational numbers. These are numbers like: $1,\ -0.1,\ \frac{3}{4},\ 15.82,\ \sqrt{2},\ -\pi$, etc.

Absolute value of real numbers

The absolute value of a real number $x$, written $|x|$, is the non-negative value of $x$.

$$|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$

Examples and step-by-step solutions

Example 1: $|7|$ Since 7 is positive: $|7| = 7$ Example 2: $|-9|$ Since -9 is negative: $|-9| = -(-9) = 9$ Example 3: Solve $|x| = 6$

$$\text{Either } x = 6 \quad \text{or} \quad x = -6$$

Final Answer: $x = \pm 6$ Example 4: Solve $|x - 3| = 2$ Case 1: $x - 3 = 2 \Rightarrow x = 5$ Case 2: $x - 3 = -2 \Rightarrow x = 1$ Final Answer: $x = 1 \text{ or } x = 5$ Example 5: Solve $|2x + 1| = 7$ Case 1: $2x + 1 = 7 \Rightarrow 2x = 6 \Rightarrow x = 3$ Case 2: $2x + 1 = -7 \Rightarrow 2x = -8 \Rightarrow x = -4$ Final Answer: $x = 3 \text{ or } x = -4$ Example 6: Solve $|x + 5| = 0$ Since absolute value is zero only when the inside is zero: $x + 5 = 0 \Rightarrow x = -5$ Final Answer: $x = -5$