Problem 29 - Algebra A Combined Function

Question

Solve each inequality. Write the solution set in interval notation. $$ x^{2}>x $$

Step-by-Step Solution

Answer

The solution is \((-\infty, 0) \cup (1, \infty)\).

Step 1: Move to One Side

Start by moving all terms to one side of the inequality to form a quadratic inequality. Subtract \( x \) from both sides to get: \[ x^2 - x > 0 \]

Step 2: Factor the Quadratic

Factor the expression on the left-hand side of the inequality. The inequality \( x^2 - x > 0 \) factors to: \[ x(x - 1) > 0 \]

Step 3: Find the Critical Points

Determine the critical points by setting each factor equal to zero. Solve \( x = 0 \) and \( x - 1 = 0 \), which gives the critical points \( x = 0 \) and \( x = 1 \).

Step 4: Test Intervals

Use the critical points to divide the number line into intervals: - \((-\infty, 0)\)- \((0, 1)\)- \((1, \infty)\)Test a point from each interval to see where the inequality holds.For \((-\infty, 0)\), choose \(x = -1\):\(-1(-1 - 1) = 2\), which is positive, so it satisfies the inequality.For \((0, 1)\), choose \(x = 0.5\): \(0.5(0.5 - 1) = -0.25\), which is negative, so it does not satisfy the inequality.For \((1, \infty)\), choose \(x = 2\): \(2(2 - 1) = 2\), which is positive, so it satisfies the inequality.

Step 5: Write the Solution in Interval Notation

The solution is where the inequality holds. This happens in the intervals \((-\infty, 0)\) and \((1, \infty)\). Therefore, the solution in interval notation is: \[ (-\infty, 0) \cup (1, \infty) \]

Key Concepts

Quadratic Functions

Quadratic functions are expressions of the form \( ax^2 + bx + c \) where \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \). These functions graph as parabolas, which can open upwards or downwards depending on the sign of \( a \). If \( a \) is positive, the parabola opens upwards, and if \( a \) is negative, it opens downwards.
In the world of quadratic inequalities, we are interested in when these functions are greater than or less than zero. This means we want to know when the parabola is entirely above or below a certain line, usually the x-axis.
Understanding the nature of quadratic functions is key to solving inequalities like \( x^2 > x \). By seeing it as \( x^2 - x > 0 \), we can work on identifying where our quadratic function's output will be greater than zero through techniques like factoring or analyzing the graph.

Critical Points

Critical points are essential values that help us determine where a quadratic inequality changes its sign or behavior. They are found by setting each factor of our quadratic expression equal to zero and solving for \( x \).
In our exercise, after factoring \( x^2 - x \) into \( x(x - 1) \), setting each part equal to zero gives us the equations \( x = 0 \) and \( x - 1 = 0 \). Solving these, we get the critical points \( x = 0 \) and \( x = 1 \).
The critical points divide our number line into distinct intervals to test. This allows us to analyze each segment to see whether the quadratic holds true under the inequality condition, giving a clearer understanding of where the solutions lie.

Interval Notation

Interval notation is a shorthand used to describe sets of numbers bounded within intervals. It is particularly useful when expressing solutions to inequalities.
There are generally three points of interest when using interval notation:

• Parentheses \(( )\) indicate that an endpoint is not included in the interval.
• Brackets \([ ]\) indicate that an endpoint is included in the interval.
• Union \( \cup \) is used to combine multiple intervals that form part of a solution.

In our quadratic inequality \( x^2 - x > 0 \), the solution is written as \((-\infty, 0) \cup (1, \infty)\). This means all numbers less than 0 and greater than 1 satisfy the inequality, while 0 and 1 themselves do not.
Interval notation provides a neat, compact way to convey solutions and handle potentially infinite solutions efficiently.

Factoring Expressions

Factoring is a fundamental process in algebra that simplifies expressions, making it easier to solve quadratic equations and inequalities. The goal is to rewrite a quadratic function as a product of simpler expressions.
With the expression \( x^2 - x \), we notice that \( x \) is a common factor, allowing us to write it as \( x(x - 1) \). Each factor represents a linear equation that when multiplied outputs the original quadratic expression.
Factoring plays a crucial role when solving inequalities because it highlights the critical points, where the expression equals zero. Once factored, the quadratic becomes more manageable, and we merely focus on solving for each factor separately.
To sum up, factoring transforms complex problems into straightforward ones by revealing underlying structures within quadratic expressions.