Question

The values of \(x\) for which \(-2 x-4 \leq(x+2)^{2} \leq-2 x-1\) is satisfied are (1) \([-5,-1]\). (2) \([-5,0]\) (3) \([-5,-4] \cup[-2,-1]\). (4) \([-5,-4] \cup[-2,-1] .\)

Short answer

Question: Determine the values of x that satisfy the inequality $$-2x-4 \leq (x+2)^2 \leq -2x-1$$. Answer: The values for \(x\) that satisfy the given inequality are \([-5, -4] \cup [-2, -1]\).

Step-by-step solution

Break down the inequality

Split the given inequality into two individual inequalities: - Left Inequality: $$-2x-4 \leq (x+2)^2$$ - Right Inequality: $$(x+2)^2 \leq -2x-1$$

Solve the left inequality

Solve the inequality $$-2x-4 \leq (x+2)^2$$: 1. Expand the square term: $$-2x-4 \leq x^2+4x+4$$ 2. Move all terms to one side to form a quadratic inequality: $$x^2+6x+8 \geq 0$$ 3. Factorize the quadratic expression: $$(x+2)(x+4) \geq 0$$ 4. Now, analyze the intervals of the roots: - \((-\infty, -4)\): Both factors are negative. The inequality doesn't hold. - \((-4, -2)\): One factor is positive and one is negative. The inequality holds. - \((-2, \infty)\): Both factors are positive. The inequality doesn't hold. Hence, the solution for the left inequality is $$x \in [-4, -2]$$.

Solve the right inequality

Solve the inequality $$(x+2)^2 \leq -2x-1$$: 1. Expand the square term: $$x^2+4x+4 \leq -2x-1$$ 2. Move all terms to one side to form a quadratic inequality: $$x^2+6x+5 \leq 0$$ 3. Factorize the quadratic expression: $$(x+1)(x+5) \leq 0$$ 4. Now, analyze the intervals of the roots: - \((-\infty, -5)\): Both factors are negative. The inequality doesn't hold. - \((-5, -1)\): One factor is positive and one is negative. The inequality holds. - \((-1, \infty)\): Both factors are positive. The inequality doesn't hold. Hence, the solution for the right inequality is $$x \in [-5, -1]$$.

Combine solutions for left and right inequalities

Now, we can find the set of values of \(x\) that satisfy both the left and right inequalities: Intersection: $$[-4, -2] \cap [-5, -1] = [-4, -2] \cup [-5, -1] = [-5, -4] \cup [-2, -1]$$ Thus, the values of \(x\) for which \(-2x-4 \leq (x+2)^2 \leq -2x-1\) is satisfied are \([-5,-4] \cup [-2,-1]\), which corresponds to option (3).

Key concepts

Quadratic Inequality

Quadratic inequalities can seem tricky, but they're just like regular inequalities with a squared term. They involve expressions like \(ax^2 + bx + c\) and you're looking to find where these expressions are greater than, less than, or equal to zero.
When faced with a quadratic inequality, first rearrange all terms from one side to the other to set up the problem. This helps convert the inequality into a more familiar quadratic equation.

• Example: For the inequality \(-2x-4 \leq (x+2)^2\), expand and move all terms to one side, giving \(x^2+6x+8 \geq 0\).

From here, solving involves factorization and interval testing, steps we'll explore further in subsequent sections.
The aim is to determine the set of \(x\) values that satisfy the inequality. Once mastered, quadratic inequalities become powerful tools for understanding polynomial behavior over ranges!

Factorization

Factorization is a crucial technique in solving quadratic inequalities, where you express the quadratic equation as a product of its factors.

• For \(x^2+6x+8 \geq 0\), factorizing gives \((x+2)(x+4) \geq 0\).
• For \(x^2+6x+5 \leq 0\), it becomes \((x+1)(x+5) \leq 0\).

By breaking down the equation into these factorized forms, it becomes easier to determine where the expression is zero and changes signs.
To solve, find the roots (or zeroes) from the factors, i.e., \(x = -2, -4\) and \(x = -1, -5\) respectively.
These roots break the number line into intervals. These intervals are key for testing where the inequality holds true.
Factorization simplifies solving by providing clear, actionable steps and insights into the polynomial's behavior across its domain.

Interval Analysis

Interval analysis involves exploring the intervals around the roots you've found.
Once the quadratic is factorized, these roots separate the number line into different segments. Each segment can be tested to see if the inequality holds within that range.

• Consider \((x+2)(x+4) \geq 0\). The roots are \(-4\) and \(-2\), creating intervals: \((-\infty, -4)\), \((-4, -2)\), \((-2, \infty)\).
• For \((x+1)(x+5) \leq 0\), the roots are \(-5\) and \(-1\), splitting into: \((-\infty, -5)\), \((-5, -1)\), \((-1, \infty)\).

Within each interval, pick a test point and substitute it into the factorized form to check if the inequality is satisfied.
Create a number line, plot the roots, and use test points from each interval for quick determination of where the inequality holds.
Interval analysis is a methodical way to see how the expression behaves across the entire number line.

Intersection of Sets

The intersection of sets in this context refers to finding common solutions between two inequalities.

• The left inequality solved for \([-4, -2]\).
• The right inequality solved for \([-5, -1]\).

To satisfy both inequalities simultaneously, identify where these solutions overlap. This "intersection" gives us the final solution.
For our example, the intersection of \([-4, -2]\) with \([-5, -1]\) results in overlapping sets: \([-5, -4]\) and \([-2, -1]\).
This is the solution for the entire quadratic inequality system.
Visually, imagine these as ranges on a number line. The intersecting portions are our solution, highlighting the exact values that meet both equations.
Understanding intersections is vital, ensuring that the final set of solutions actually meets all conditions outlined by the problem.