Solve the Inequality for x 1< =|x|< =4

Solution:

Step 1

Determine the set of values for $x$ that satisfy $1\le |x|$.

Step 1.1

Reformulate the inequality with $x$ on the left: $|x|\ge 1$.

Step 1.2

Express $|x|\ge 1$ using a piecewise definition.

Step 1.2.1

Identify the range where $x$ is positive or zero: $x\ge 0$.

Step 1.2.2

For positive $x$, the inequality simplifies to: $x\ge 1$.

Step 1.2.3

Identify the range where $x$ is negative: $x<0$.

Step 1.2.4

For negative $x$, the inequality becomes: $-x\ge 1$.

Step 1.2.5

Combine the conditions into a piecewise format:

$$\{\begin{array}{ll}x\ge 1& \text{if }x\ge 0\\ -x\ge 1& \text{if }x<0\end{array}$$

Step 1.3

The solution for $x\ge 0$ is: $x\ge 1$.

Step 1.4

Solve the inequality $-x\ge 1$ by dividing by $-1$.

Step 1.4.1

Invert the inequality sign upon dividing by a negative number: $x\le -1$.

Step 1.4.2

The left side simplifies directly.

Step 1.4.2.1

Negative divided by negative is positive: $x\le -1$.

Step 1.4.2.2

Simplify the division: $x\le -1$.

Step 1.4.3

The right side simplifies to: $x\le -1$.

Step 1.5

Combine the solutions: $x\le -1$ or $x\ge 1$.

Step 2

Identify the set of values for $x$ that fulfill $|x|\le 4$.

Step 2.1

Express $|x|\le 4$ using a piecewise definition.

Step 2.1.1

For non-negative $x$: $x\ge 0$.

Step 2.1.2

For positive $x$, the inequality simplifies to: $x\le 4$.

Step 2.1.3

For negative $x$: $x<0$.

Step 2.1.4

For negative $x$, the inequality becomes: $-x\le 4$.

Step 2.1.5

Combine the conditions into a piecewise format:

$$\{\begin{array}{ll}x\le 4& \text{if }x\ge 0\\ -x\le 4& \text{if }x<0\end{array}$$

Step 2.2

The solution for $x\ge 0$ is: $0\le x\le 4$.

Step 2.3

Solve the inequality $-x\le 4$ for negative $x$.

Step 2.3.1

Divide by $-1$ and reverse the inequality: $x\ge -4$.

Step 2.3.1.1

The left side simplifies directly.

Step 2.3.1.2

Negative divided by negative is positive: $x\ge -4$.

Step 2.3.1.3

The right side simplifies to: $x\ge -4$.

Step 2.4

Combine the solutions: $-4\le x\le 4$.

Step 3

The overall solution is the intersection of the intervals from Steps 1 and 2.

Step 4

Determine the intersection: $-4\le x\le -1$ or $1\le x\le 4$.

Step 5

The solution can be represented in different notations.

Inequality Form: $-4\le x\le -1$ or $1\le x\le 4$ Interval Notation: $[-4,-1]\cup [1,4]$

Knowledge Notes:

To solve the inequality $1\le |x|\le 4$, we need to consider the definition of absolute value, which is the distance of a number from zero on the number line, without considering direction. The absolute value of $x$, denoted as $|x|$, is defined as:

$$|x|=\{\begin{array}{ll}x& \text{if }x\ge 0\\ -x& \text{if }x<0\end{array}$$

When solving absolute value inequalities, we consider two cases: one where the expression inside the absolute value is non-negative, and one where it is negative. We then combine the solutions from both cases.

For inequalities, it's important to remember that when we multiply or divide both sides by a negative number, we must reverse the direction of the inequality sign.

The final solution is expressed as the intersection of the solutions for the two inequalities, which can be represented in inequality form or interval notation. Interval notation is a way of writing subsets of the real number line. An interval is written with brackets or parentheses, where square brackets [ ] denote inclusive boundaries and parentheses ( ) denote exclusive boundaries.