Question

Solve the inequality: \(\frac{x-1}{x+4}>0\)

Short answer

The solution is \ x \text{ in } (-\boldsymbol\infty, -4) \text{ or } (1, \boldsymbol\infty) \.

Step-by-step solution

Identify the critical points

Set the numerator and denominator of the inequality \(\frac{x-1}{x+4}>0\) equal to zero. These points are where the expression is undefined or equal to zero. \[ x - 1 = 0 \rightarrow x = 1 \ x + 4 = 0 \rightarrow x = -4 \] Thus, the critical points are \ x = 1 \ and \ x = -4 \.

Determine the intervals

Use the critical points to divide the number line into intervals: \(-\boldsymbol\infty, -4 \), \(-4, 1 \), and \(1, \boldsymbol\infty \).

Test a point from each interval

To determine the sign of the expression \(\frac{x-1}{x+4}\) in each interval, choose a test point within each interval and substitute it into the expression. \[ \text{Interval } (-\boldsymbol\infty, -4) \text{, test with } x = -5 \rightarrow \frac{-5-1}{-5+4} = \frac{-6}{-1} = 6 > 0 \text{Interval } (-4, 1) \text{, test with } x = 0 \rightarrow \frac{0-1}{0+4} = \frac{-1}{4} = -\frac{1}{4} < 0 \text{Interval } (1, \boldsymbol\infty) \text{, test with } x = 2 \rightarrow \frac{2-1}{2+4} = \frac{1}{6} > 0 \] Therefore, the expression is positive in the intervals \[ (-\boldsymbol\infty, -4) \text{ and } (1, \boldsymbol\infty) \] and negative in the interval \((-4, 1)\).

Identify the solution

Since we want the inequality \(\frac{x-1}{x+4} > 0 \), include the intervals where the expression is positive. Also, exclude the critical points where the expression is zero or undefined. \[ x \text{ in } (-\boldsymbol\infty, -4) \text{ or } (1, \boldsymbol\infty) \]

Key concepts

critical points

In algebra, critical points are the values of the variable that make the numerator or denominator of an expression zero. For rational inequalities like \(\frac{x-1}{x+4} > 0\), identifying critical points is important because they help in breaking down the number line into manageable intervals.

Here's a simplified approach:

• Set the numerator equal to zero: \(x-1=0\rightarrow x=1\). This is a point where the expression itself is zero.
• Set the denominator equal to zero: \(x+4=0\rightarrow x=-4\). This is a point where the expression is undefined.

Therefore, our critical points are at \(x=1\) and \(x=-4\). These points are key to determining the behaviour of the inequality.

number line intervals

After finding the critical points, the next step is to divide the entire number line into segments or intervals.

Here's how to do it:

• The number line extends from \(-\boldsymbol\textINF\) to \( \boldsymbol\textINF \).
• At each critical point, divide the number line into sections. For \(x=1\) and \(x=-4\), we get three intervals: \((-\boldsymbol\textINF,-4)\), \((-4,1)\), and \((1, \boldsymbol\textINF)\).

These intervals are necessary to test the sign of the expression in each section and find where the inequality holds true.

test points

Selecting test points is crucial for determining the sign of the rational expression within each interval.

Let's break it down:

• Choose any value within each interval.
• Substitute the test point into the expression to see whether the result is positive or negative.

For example:

• For \((-\boldsymbol\textINF, -4)\), use \(x = -5\). Substituting, we get \(\frac{-5-1}{-5+4} = \frac{-6}{-1} = 6 > 0\).
• For \((-4,1)\), use \(x=0\). Substituting, we get \(\frac{0-1}{0+4} = \frac{-1}{4} = -\frac{1}{4} < 0\).
• For \((1, \boldsymbol\textINF)\), use \(x=2\). Substituting, we get \(\frac{2-1}{2+4} = \frac{1}{6} > 0\).

These test points help you determine if the expression \( \frac{x-1}{x+4}\) is positive or negative within each interval.

rational inequalities

Rational inequalities involve expressions that are ratios of polynomials. Solving them requires determining where the expression is greater than or less than zero.
Here’s how:

• Identify the critical points by setting the numerator and denominator equal to zero.
• Divide the number line into intervals based on these critical points.
• Use test points from each interval to check the sign of the expression.
• Determine which intervals satisfy the inequality.

For the inequality \( \frac{x-1}{x+4} > 0\), the expression is positive in the intervals \((-\boldsymbol\textINF, -4)\) and \((1, \boldsymbol\textINF)\), and thus our solution is:
\[ x \text{ in } (-\boldsymbol\textINF, -4) \text{ or } (1, \boldsymbol\textINF) \].
It’s essential to exclude the critical points where the expression is zero or undefined, to avoid division by zero errors.