Solve the Inequality for x -7|x+12|< =-14
The problem presents an inequality to solve for the variable x. The inequality involves an absolute value expression of x, which is being subtracted by 7 times the absolute value of x+12. The inequality is less than or equal to negative fourteen (-14). The task here would be to find the range of values for x that satisfy this inequality. A solution would typically involve considering the definition of the absolute value function, which means splitting the problem into the case where (x+12) is non-negative and the case where (x+12) is negative. Solving the inequality for x will then require handling each case separately.
$- 7 \left|\right. x + 12 \left|\right. \leq - 14$
Answer
Solution:
Step 1: Convert the inequality into a piecewise expression
Step 1.1: Identify when $x + 12$ is non-negative: $x + 12 \geq 0$
Step 1.2: Isolate $x$: $x \geq -12$
Step 1.3: For $x + 12 \geq 0$, the absolute value is not needed: $-7(x + 12) \leq -14$
Step 1.4: Identify when $x + 12$ is negative: $x + 12 < 0$
Step 1.5: Isolate $x$: $x < -12$
Step 1.6: For $x + 12 < 0$, consider the absolute value as negation: $-7(-x - 12) \leq -14$
Step 1.7: Combine both cases into a piecewise inequality:
- $-7(x + 12) \leq -14$ for $x \geq -12$
- $-7(-x - 12) \leq -14$ for $x < -12$
Step 1.8: Simplify the first case: $-7x - 84 \leq -14$
Step 1.9: Simplify the second case: $7x + 84 \leq -14$
Step 2: Solve the first case $-7x - 84 \leq -14$
Step 2.1: Move constants to one side: Add $84$ to both sides.
Step 2.2: Divide by $-7$ and reverse inequality: $x \geq 10$
Step 3: Solve the second case $7x + 84 \leq -14$
Step 3.1: Move constants to one side: Subtract $84$ from both sides.
Step 3.2: Divide by $7$: $x \leq -14$
Step 4: Combine the solutions
- The solution is $x \leq -14$ or $x \geq 10$
Step 5: Express the solution in different forms
- Inequality Form: $x \leq -14$ or $x \geq 10$
- Interval Notation: $(-\infty, -14] \cup [10, \infty)$
Step 6: End of the solution process
Knowledge Notes:
Absolute Value: The absolute value of a number is its distance from zero on the number line, without considering direction. For any real number $a$, $|a| = a$ if $a \geq 0$ and $|a| = -a$ if $a < 0$.
Piecewise Functions: A piecewise function is defined by different expressions based on different intervals of the input variable.
Inequalities: When solving inequalities, if you multiply or divide by a negative number, you must reverse the direction of the inequality sign.
Interval Notation: This is a way of writing subsets of the real number line. An interval includes all numbers between two endpoints. It can be inclusive, denoted by square brackets [ ], or exclusive, denoted by parentheses ( ).
Combining Inequalities: When two inequalities have no overlap, their solutions can be combined using the word "or" to indicate that values satisfying either inequality are part of the solution set.
