Question

Solve the inequality algebraically. \(\frac{1}{2} x^2-x>4\)

Short answer

The solution to the inequality is \(x<-2\) or \(x>4\).

Step-by-step solution

Arrange the inequality

First, bring all terms to one side of the inequality to form a quadratic inequality \(\frac{1}{2} x^2 - x - 4 > 0\).

Simplify the inequality

To make calculations easier, multiply the entire inequality by 2, the denominator of 1/2, which transforms the inequality into \(x^2 - 2x - 8 > 0\).

Solve for roots of corresponding equation

Now, solve the related quadratic equation \(x^2 - 2x - 8 = 0\). This equation can be factored into \((x-4)(x+2) = 0\). From this we find the roots \(x=4\) and \(x=-2\).

Find the solution intervals

We use these roots to divide the number line into three parts: \(x<-2\), \(-24\). Then substitute a number from each part into the inequality to determine where the solutions exist. By checking, we find the inequality holds when \(x<-2\) and \(x>4\).

Key concepts

Quadratic Equations

Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0\), where "a," "b," and "c" are constants, and "x" represents an unknown. They are called "quadratic" because they involve terms up to \(x^2\), representing the second degree. The standard form of a quadratic equation allows us to examine relationships, solve for unknowns, and apply various methods such as factoring, completing the square or using the quadratic formula. Quadratic equations can have 0, 1, or 2 real roots depending on the discriminant \(b^2-4ac\).

Understanding the importance of each coefficient helps analyze the graph of the equation (a parabola), where "a" affects the direction and width, "b" affects the symmetry, and "c" the vertical shift.

Number Line Analysis

Number line analysis is a visual tool used to solve inequalities and understand solution sets. It involves placing critical points, often derived from setting the expression of an inequality to zero and solving, onto a number line. The number line is then divided into sections based on these critical points. In this process, each section represents different potential solutions.

The effectiveness of number line analysis lies in its ability to quickly test which sections satisfy the original inequality by picking test points. For example, with roots at \(x = -2\) and \(x = 4\), the number line is divided into intervals: \(x < -2\), \(-2 < x < 4\), and \(x > 4\). By substituting test values from each interval back into the inequality, one can determine which intervals contain valid solutions.

Factoring Quadratics

Factoring is a crucial method for solving quadratic equations. It involves writing the quadratic equation as a product of two binomials. For example, \(x^2 - 2x - 8\) can be factored into \((x-4)(x+2) = 0\). This factorization reveals the equation's roots, which are the values of \(x\) that make the equation equal to zero.

Factoring requires identifying two numbers that multiply to give the constant term (here, -8) and add to give the coefficient of the middle term (here, -2). Recognizing these pairs allows easy breakdown of the quadratic, thus providing an efficient route to derive critical points for inequalities.

Inequality Solutions

When solving inequalities, the goal is to find the set of values that make the inequality true. After factoring the quadratic and identifying the roots, the solution to the inequality involves further analysis. Specifically, once critical points are defined, it must be determined which intervals satisfy the inequality \(x^2 - 2x - 8 > 0\).

For the inequality solution:

• Test values from each number line interval.
• Substituting \(x=-3\) (from \(x<-2\)) into the inequality returns a true condition.
• Similarly, \(x=0\) (from \(-24\)) does.

These verifications confirm the solution intervals, resulting in: \(x < -2\) and \(x > 4\). Geometry and number line divisions ensure that all potential solutions are considered.