Solve the Inequality for x -7|x+12|< =-14

Solution:

Step 1: Convert the inequality into a piecewise expression

• Step 1.1: Identify when $x+12$ is non-negative: $x+12\ge 0$

• Step 1.2: Isolate $x$: $x\ge -12$

• Step 1.3: For $x+12\ge 0$, the absolute value is not needed: $-7(x+12)\le -14$

• 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: $-7(-x-12)\le -14$

• Step 1.7: Combine both cases into a piecewise inequality:

  • $-7(x+12)\le -14$ for $x\ge -12$
  • $-7(-x-12)\le -14$ for $x<-12$

• Step 1.8: Simplify the first case: $-7x-84\le -14$

• Step 1.9: Simplify the second case: $7x+84\le -14$

Step 2: Solve the first case $-7x-84\le -14$

• Step 2.1: Move constants to one side: Add $84$ to both sides.

• Step 2.2: Divide by $-7$ and reverse inequality: $x\ge 10$

Step 3: Solve the second case $7x+84\le -14$

• Step 3.1: Move constants to one side: Subtract $84$ from both sides.

• Step 3.2: Divide by $7$: $x\le -14$

Step 4: Combine the solutions

• The solution is $x\le -14$ or $x\ge 10$

Step 5: Express the solution in different forms

• Inequality Form: $x\le -14$ or $x\ge 10$
• Interval Notation: $(-\mathrm{\infty },-14]\cup [10,\mathrm{\infty })$

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 $a\ge 0$ 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.