Question

Solve the inequality. Then graph the solution set on the real number line. \(x^{2}+4 x+4 \geq 9\)

Short answer

The solution set of the inequality \(x^{2}+4 x+4 \geq 9\) is x in (-∞,-5] union [1, ∞), shown on the number line with filled dots at -5 and 1, and arrows towards ∞ from 1 and -5 showing all values up to -∞ and down to ∞ are part of the solution.

Step-by-step solution

Rewrite the inequality

First, subtract 9 from both sides of the inequality to set up the equation in a more familiar format. This comes out to form a quadratic inequality, which then becomes \(x^2+4x-5\geq0\)

Factorize the quadratic expression

The quadratic expression can be written in the form of \((x-a)(x-b)\). We need to find such 'a' and 'b' which on multiplication give '-5' and on addition give '4'. Hence, factor the equation as \((x+5)(x-1)\geq0\)

Find the roots

Then, set each factor to be zero and solve for 'x'. Thus, the roots of the inequality are x=-5 and x=1

Identify solution sets

Because our original inequality is 'greater than or equal to', the solutions must be both the roots and the values of 'x' that make \((x+5)(x-1)\) positive. So, x is in (-∞,-5] union [1, ∞). Means x is less than or equal to -5 and greater than or equal to 1.

Graph the solution set

On the real number line, put closed dots on -5 and 1, and draw arrows pointing to the right from 1 and to the left from -5. The area in between (-5,1) is unshaded indicating that it isn't part of the solution

Key concepts

Factoring Quadratics

Factoring quadratics is a technique used to simplify and solve quadratic expressions. A quadratic expression is typically in the form of \( ax^2 + bx + c \). The goal of factoring is to express the quadratic equation as the product of two linear expressions. This makes it easier to find the roots or solutions of the equation.

• First, identify the coefficients: \( a \), \( b \), and \( c \).
• Next, find two numbers that multiply to \( a \, c \) (product) and add up to \( b \) (sum).
• Rewrite the quadratic expression using these numbers and split the middle term based on the sum.
• Factor by grouping, if necessary, to express as a product of two binomials.

This step is crucial in solving polynomial inequalities because it reveals the roots—values that make the expression equal to zero. Knowing these helps in understanding the intervals to test when solving the inequality.

Real Number Line Graphing

Graphing on a real number line is a visual way to represent the solution set of an inequality. After factoring the quadratic expression and finding its roots, you move to the graphical representation.
Start by placing the identified roots on the number line. For the inequality \((x+5)(x-1) \geq 0\), the roots are \(-5\) and \(1\). Here is how you proceed:

• Draw the number line and mark the roots with symbols indicating inclusion or exclusion ([ ] for inclusion, ( ) for exclusion).
• A closed dot or bracket is used at \(-5\) and \(1\) because the inequality is "greater than or equal to." These points include the values \(-5\) and \(1\).
• Shade the direction of the inequality where the values satisfy \((x+5)(x-1) \geq 0\). This results in shading to the left of \(-5\) and to the right of \(1\).
• Leave unshaded the interval between the roots, as it doesn't meet the inequality.

Real number line graphing allows students to visually understand solutions, simplifying the complexity of abstract algebraic concepts.

Solving Polynomial Inequalities

Solving polynomial inequalities involves determining the set of real numbers that satisfy the inequality. Once you have factored the quadratic and identified the roots, the next step is to decide the intervals on the number line that satisfy the inequality expression.
To solve the inequality \((x+5)(x-1) \geq 0\):

• Determine the roots of the quadratic, \(-5\) and \(1\).
• Divide the number line into sections based on these roots: \((-\infty, -5]\), \((-5, 1)\), and \([1, \infty)\).
• Test a sample value from each section in the inequality to check if it satisfies \((x+5)(x-1) \geq 0\).
• The sections that return true indicate where the inequality holds. For this problem, these are \((-\infty, -5]\) and \([1, \infty)\).

This step-by-step testing helps build a clear picture of the solution set, ensuring that no valid intervals are missed. It is a practical way to understand where the function lies above or below the line on the graph, which is essential for accurate graphing.