Solve the Absolute Value Inequality for x 15=|10-3x|
The problem presented is an absolute value inequality, which involves solving for the variable \( x \) within the context of an inequality that contains an absolute value expression. The expression at hand is \( 15 = |10 - 3x| \). The goal is to determine the values of \( x \) for which this inequality holds true. To solve this, one would typically break it down into two separate scenarios: one where the contents of the absolute value are positive and one where they are negative, ultimately finding a solution set for \( x \).
$15 = \left|\right. 10 - 3 x \left|\right.$
Answer
Solution:
Step 1:
Express the equation without the absolute value, denoted as $|10 - 3x| = 15$.
Step 2:
Eliminate the absolute value, introducing a plus/minus sign on the equation's right side, since $|x| = \pm x$.
$$10 - 3x = \pm 15$$
Step 3:
Solve for both the positive and negative scenarios separately.
Step 3.1:
Start with the positive scenario.
$$10 - 3x = 15$$
Step 3.2:
Isolate the variable $x$ by moving other terms to the opposite side.
Step 3.2.1:
Subtract 10 from both sides.
$$-3x = 15 - 10$$
Step 3.2.2:
Perform the subtraction.
$$-3x = 5$$
Step 3.3:
Divide by the coefficient of $x$ to solve for $x$.
Step 3.3.1:
Divide both sides by $-3$.
$$\frac{-3x}{-3} = \frac{5}{-3}$$
Step 3.3.2:
Simplify the equation.
Step 3.3.2.1:
Reduce the common factors.
$$x = \frac{5}{-3}$$
Step 3.3.3:
Rearrange the negative sign.
$$x = -\frac{5}{3}$$
Step 3.4:
Now, address the negative scenario.
$$10 - 3x = -15$$
Step 3.5:
Again, isolate the variable $x$.
Step 3.5.1:
Subtract 10 from both sides.
$$-3x = -15 - 10$$
Step 3.5.2:
Perform the subtraction.
$$-3x = -25$$
Step 3.6:
Divide by the coefficient of $x$ to find the solution.
Step 3.6.1:
Divide both sides by $-3$.
$$\frac{-3x}{-3} = \frac{-25}{-3}$$
Step 3.6.2:
Simplify the equation.
Step 3.6.2.1:
Reduce the common factors.
$$x = \frac{-25}{-3}$$
Step 3.6.3:
Recognize that a negative divided by a negative is positive.
$$x = \frac{25}{3}$$
Step 3.7:
Combine the solutions from the positive and negative scenarios.
$$x = -\frac{5}{3}, \frac{25}{3}$$
Step 4:
Present the solution in various forms.
Exact Form:
$$x = -\frac{5}{3}, \frac{25}{3}$$
Decimal Form:
$$x = -1.666..., 8.333...$$
Mixed Number Form:
$$x = -1\frac{2}{3}, 8\frac{1}{3}$$
Knowledge Notes:
To solve an absolute value inequality such as $|10 - 3x| = 15$, one must understand the definition of absolute value, which is the distance a number is from zero on the number line, without considering direction. The absolute value of a number is always nonnegative.
When you remove the absolute value from an equation, you must consider both the positive and negative possibilities because $|x| = x$ if $x$ is positive or zero, and $|x| = -x$ if $x$ is negative.
The process of solving the equation involves isolating the variable on one side. This typically requires moving terms that do not contain the variable to the other side of the equation and then simplifying.
When dividing both sides of an equation by a negative number, the inequality direction remains the same since we are dealing with an equation, not an inequality. However, the sign of the number itself changes.
The final solution of an absolute value equation can be expressed in different forms: exact form (as a fraction), decimal form, or mixed number form. It is important to be comfortable with converting between these forms for clarity and understanding.
