Question

Solve each inequality algebraically. $$\frac{x-4}{2 x+4} \geq 1$$

Short answer

-8 < x < -2

Step-by-step solution

- Isolate the Fraction

To solve the inequality \( \frac{x-4}{2x+4} \geq 1 \), first subtract 1 from both sides to isolate the fraction: \[ \frac{x-4}{2x+4} - 1 \geq 0 \]

- Common Denominator

Combine the terms on the left side by expressing 1 as a fraction with a common denominator: \[ \frac{x-4}{2x+4} - \frac{2x+4}{2x+4} \geq 0 \]

- Simplify the Expression

Combine the fractions over the common denominator and simplify: \[ \frac{(x-4) - (2x+4)}{2x+4} \geq 0 \] Simplify the numerator: \[ \frac{x-4-2x-4}{2x+4} \geq 0 \] \[ \frac{-x-8}{2x+4} \geq 0 \]

- Critical Points

Find the critical points by setting the numerator and the denominator to zero separately. For the numerator: \[-x-8 = 0 \implies x = -8 \] For the denominator: \[2x+4 = 0 \implies x = -2 \]

- Test Intervals

Test intervals around the critical points \( x = -8 \) and \( x = -2 \). The intervals to consider are: \((-∞, -8)\), \((-8, -2)\), and \((-2, ∞)\). Check the sign of \( \frac{-x-8}{2x+4} \) within these intervals by selecting test points.

- Interval Analysis

For \( x < -8 \) (e.g., \( x = -10 \)): \[ \frac{-(-10)-8}{2(-10)+4} = \frac{18}{-16} < 0 \] For \( -8 < x < -2 \) (e.g., \( x = -5 \)): \[ \frac{-(-5)-8}{2(-5)+4} = \frac{-3}{-6} > 0 \] For \( x > -2 \) (e.g., \( x = 0 \)): \[ \frac{-0-8}{2(0)+4} = \frac{-8}{4} < 0 \] The inequality \( \frac{-x-8}{2x+4} \geq 0 \) is satisfied where the numerator and denominator are not zero and their product is non-negative.

- Solution Set

Combine the valid intervals. The solution to the inequality is: \[ -8 < x < -2 \]

Key concepts

Algebraic Manipulation

Algebraic manipulation is an essential skill in solving inequalities. It involves the use of basic algebraic operations such as addition, subtraction, multiplication, and division to isolate variables and simplify expressions. Here, let's break this concept down for our inequality problem:

We start with the inequality \[ \frac{x-4}{2x+4} \geq 1 \]. The goal is to isolate the fraction, so we subtract 1 from both sides: \[ \frac{x-4}{2x+4} - 1 \geq 0 \].

Next, express 1 with a common denominator: \[ \frac{x-4}{2x+4} - \frac{2x+4}{2x+4} \geq 0 \]. Then, combine the fractions over the common denominator: \[ \frac{(x-4) - (2x+4)}{2x+4} \geq 0 \]. Simplify the numerator to \[ \frac{-x-8}{2x+4} \geq 0 \].

By performing these steps, we reduced the original inequality into a simpler form that is easier to analyze.

Critical Points

Identifying critical points is vital when solving inequalities. These are values of x where the numerator or the denominator of a rational expression equals zero. In our example, to find these points, we set both the numerator and the denominator of \[ \frac{-x-8}{2x+4} \geq 0 \] to zero:

• For the numerator: \[-x-8 = 0 \implies x = -8 \]
• For the denominator: \[2x+4 = 0 \implies x = -2 \]

Thus, our critical points are \[ x = -8 \] and \[ x = -2 \]. These points divide the number line into intervals, which we will test next.

Interval Testing

Interval testing helps determine where the inequality holds true. We use the critical points to split the number line into distinct intervals. For our problem, the critical points \[ x = -8 \] and \[ x = -2 \] give us three intervals to test: \[ (-fty, -8) \], \[ (-8, -2) \], and \[ (-2, fty) \].

We pick a test point from each interval and substitute it back into the simplified inequality \[ \frac{-x-8}{2x+4} \geq 0 \]:

• For \[ x < -8 \] (e.g., \[ x = -10 \]): \[\frac{18}{-16} < 0 \]
• For \[ -8 < x < -2 \] (e.g., \[ x = -5 \]): \[\frac{-3}{-6} > 0 \]
• For \[ x > -2 \] (e.g., \[ x = 0 \]): \[\frac{-8}{4} < 0 \]

From this, we see the inequality holds only in the interval \[ -8 < x < -2 \].

Rational Inequalities

Rational inequalities involve fractions whose numerators and denominators are polynomials. Solving them typically involves:

• Isolating the rational expression
• Finding the critical points
• Testing intervals determined by these points

In our example, we had the rational inequality \[ \frac{x-4}{2x+4} \geq 1 \]. After isolating and simplifying it, we identified \[ x = -8 \] and \[ x = -2 \] as critical points and used interval testing to find the solution.

We conclude that the solution set for the inequality is where the numerator and denominator make the fraction non-negative, i.e., \[ -8 < x < -2 \]. This systematic method ensures that all parts of the inequality are carefully analyzed for a clear solution.