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Magazine Chapter 10: Problem 60
Solve each inequality. Write the solution set in interval notation and graph it. $$ x^{2}+4 x+4>0 $$
Short Answer
Expert verified
The solution set is \((-\infty, -2) \cup (-2, \infty)\).
Step by step solution
01
Identify the Mathematical Expression Type
The inequality given is a quadratic inequality because it involves a quadratic expression, specifically the function \(x^2 + 4x + 4\).
02
Factor the Quadratic Expression
Attempt to factor the quadratic expression \(x^2 + 4x + 4\). This can be factored as \((x + 2)^2\), since \(x^2 + 4x + 4 = (x + 2)(x + 2)\).
03
Analyze the Sign of the Expression
The inequality \((x + 2)^2 > 0\) suggests that the expression is always non-negative since it is a square. We need to determine for which values it is strictly greater than zero.
04
Identify Value Where Expression Equals Zero
The expression \((x + 2)^2\) equals zero when \(x + 2 = 0\), which is \(x = -2\). Therefore, the expression is positive for all \(x\) not equal to \(-2\).
05
Determine the Solution Set
The solution set of the inequality \((x + 2)^2 > 0\) includes all \(x\) except \(x = -2\). In interval notation, this is \((-\infty, -2) \cup (-2, \infty)\).
06
Graph the Solution Set
On a number line, draw open circles at \(-2\) to indicate that \(-2\) is not included in the solution set, and shade the lines extending from \(-\infty\) to \(-2\) and from \(-2\) to \(\infty\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Interval Notation
Interval notation is a helpful way to succinctly express a set of numbers on the number line without listing every number individually. It uses brackets and parentheses to describe intervals, with each symbol having a specific meaning. For example:
This notation helps in communicating the idea that all values except \( x = -2 \) satisfy the inequality, providing clarity and precision.
- Parentheses \( ( \, \text{or} \, ) \) are used when a number is not included in the set, often called an 'open interval.'
- Brackets \( [ \, \text{or} \, ] \) are used when a number is included, known as a 'closed interval.'
This notation helps in communicating the idea that all values except \( x = -2 \) satisfy the inequality, providing clarity and precision.
Graphing Inequalities
Graphing inequalities on a number line can transform abstract algebraic expressions into visual representations, making them easier to understand. Here鈥檚 how you can graph the inequality \( (x + 2)^2 > 0 \) effectively:
- Identify the critical points, which are the solutions where the expression equals zero. For \( (x+2)^2 \), the critical point is \( x = -2 \).
- On the number line, mark \( -2 \) with an open circle. The open circle signifies that the point is not part of the solution set (since the inequality is \( > 0 \), not \( \ge 0 \)).
- Shade the number line in two segments: from \( -\infty \) to \( -2 \) and from \( -2 \) to \( \infty \), indicating these are the values where the original inequality holds true.
Factoring Quadratic Expressions
Factoring quadratic expressions is a crucial skill in solving quadratic inequalities effectively and efficiently. A quadratic expression in the form of \( ax^2 + bx + c \) can often be rewritten as a product of binomials.
For the expression \( x^2 + 4x + 4 \), you may notice it factors neatly into \( (x + 2)^2 \). This means:
For the expression \( x^2 + 4x + 4 \), you may notice it factors neatly into \( (x + 2)^2 \). This means:
- Finding numbers that multiply to the constant term (4) and add up to the linear coefficient (4) guides the factoring process.
- Since both the factors are identical \( (x + 2) \), this quadratic is known as a 'perfect square.' Recognizing perfect squares can simplify problems, as they often relate directly to equations involving squares.
