What is an Absolute Value?The absolute value of a number is its distance from 0 on a number line, disregarding the direction. In other words, it is always a non-negative number. For example, the absolute value of both -3 and 3 is 3. It is denoted by two vertical bars, like this: |x|.
How to Solve Absolute Value Equations?To solve an absolute value equation, you need to isolate the absolute value expression and then set up two separate equations to account for the two possible values inside the absolute value bars.
Example 1: Solve |x - 5| = 31. Write two separate equations without the absolute value bars: - x - 5 = 3 - x - 5 = -3
2. Solve each equation: - For x - 5 = 3: Add 5 to both sides to get x = 8. - For x - 5 = -3: Add 5 to both sides to get x = 2.
So, the solutions are x = 8 and x = 2.
How to Solve Absolute Value Inequalities?Absolute value inequalities can come in two main forms:1. |x| < a2. |x| > a
For each form, you'd follow different steps to solve them.
Example 2: Solve |x - 4| < 61. Write the inequality without the absolute value and form two compound inequalities: - -6 < x - 4 < 6
2. Solve the compound inequalities: - Add 4 to all parts of the inequality: -6 + 4 < x - 4 + 4 < 6 + 4 - Simplify to get: -2 < x < 10
So, the solution is -2 < x < 10.
Example 3: Solve |2x + 1| > 51. Write two separate inequalities: - 2x + 1 > 5 - 2x + 1 < -5
2. Solve each inequality: - For 2x + 1 > 5: Subtract 1 from both sides to get 2x > 4, then divide by 2 to get x > 2. - For 2x + 1 < -5: Subtract 1 from both sides to get 2x < -6, then divide by 2 to get x < -3.
So, the solution is x < -3 or x > 2.
Common Mistakes To Avoid:- Forgetting to set up two equations or inequalities.- Ignoring the non-negative rule of absolute values.- Failing to flip the inequality sign when multiplying or dividing by a negative number.
By understanding and practicing these steps, you can solve absolute value equations and inequalities confidently and accurately.
Inscribed Circle A right triangle has sides of length 1 and 2 as shown in the accompanying figure. Find the exact radius of the inscribed circle. (FI…
Use a calculator to find a rational number $r$ such that $\left|r-\pi^{2}\right|<10^{-4}$.
If $2 < x < 6,$ which of the following statements about $x$ are necessarily true, and which are not necessarily true? $$ \begin{array}{ll}{\text { …
Sketch the region in the coordinate plane that satisfies both the inequalities $x^{2}+y^{2} \leq 9$ and $y \geq|x| .$ What is the area of this region?
Watch the video solution with this free unlock.
EMAIL
PASSWORD