Question

Solve the inequality. Then graph the solution set on the real number line. \(x^{2}+2 x>3\)

Short answer

The solution to the inequality \(x^{2}+2x > 3\) is x in (-\infty, -3) U (1, \infty).

Step-by-step solution

Arrange into Standard Quadratic Form

The problem can be rewritten as a quadratic inequality in standard form. Subtract 3 from both sides of the inequality to achieve this. The resulting inequality is:\(x^{2}+2x-3 > 0\)

Factorize

Factor the quadratic expression as per the factors of -3 that add up to 2. This becomes \((x-1)(x+3) > 0\)

Determine the roots

To find the roots of the quadratic equation (where \(x^{2}+2x-3=0\)), solve the equation \(x-1=0\) and \(x+3=0\) which give the roots x=1 and x=-3 respectively.

Determine the intervals

The inequality \((x-1)(x+3) > 0\), means \(x^{2}+2x-3 > 0\) is looking for values of x where the parabolic graph is above the x-axis (y>0).The roots give three intervals on the number line: \(-\infty\) to -3, -3 to 1 and 1 to \(\infty\). Test each interval with a number within that range to see whether it satisfies the inequality or not.

Graph the solution set on the number line.

Numbers from the interval (-\infty, -3) and (1, \infty) satisfy the inequality so the solution set is (-\infty, -3) U (1, \infty). Include -3 and 1 on a number line and shade in the regions that fall under the intervals that satisfy the inequality. The values of x in those regions (excluding -3 and 1) are the solutions to the inequality. Remember to use open circles as -3 and 1 are not inclusive.

Key concepts

Quadratic Inequality Graphing

Understanding how to graph quadratic inequalities is an essential skill in algebra. Let's consider the inequality from our exercise, \(x^2 + 2x > 3\). After rearranging it into standard quadratic form, \(x^2 + 2x - 3 > 0\), we get an expression that represents a parabola when set to equal zero.

Graphing this parabola involves finding its roots, where it intersects the x-axis. These roots divide the x-axis into intervals which can be tested to determine where the parabola lies above or below the x-axis. In our exercise, the roots are x = 1 and x = -3, creating three intervals: \((-\text{infty}, -3), (-3, 1)\), and \((1, \text{infty})\).

To graph the solution set on a number line, we mark the roots with open circles (since they are not included in the solution) and shade the intervals that satisfy the inequality, which in this case are \((-\text{infty}, -3)\) and \((1, \text{infty})\). The shading indicates where the values of x make the original inequality true, meaning the parabolic curve lies above the x-axis in those regions.

Factorization of Quadratic Equations

Factorization is a process that breaks down an expression into simpler multipliers that, when multiplied together, give you the original expression. In the context of a quadratic equation, such as the one from our problem, \(x^2 + 2x - 3\), we look for two binomials whose product is the quadratic equation. The coefficients of x in the binomials need to add up to the middle term of the original quadratic, and the constant term must be the product of the constants in the binomials.

For our equation, the factorized form is \((x - 1)(x + 3)\). Notice how the factors of -3 add up to +2, which satisfies our requirement for the middle term of the quadratic expression. When factorized, solving the inequality becomes straightforward, since you can analyze the individual factors to understand the intervals that make the entire expression greater than zero.

Intervals and Inequalities

Intervals and inequalities represent portions of the number line where an expression or equation may or may not hold true. An interval describes a range of numbers between two endpoints, which can be numbers or infinity symbols, indicating that the interval extends indefinitely in positive or negative directions.

When we solve quadratic inequalities, we're seeking x-values within specific intervals that make the inequality true. In our example, after finding where our factorized expression \((x - 1)(x + 3) > 0\) changes signs, we're left with the intervals \((-\text{infty}, -3), (-3, 1)\), and \((1, \text{infty})\). Selecting a test value from each interval helps determine which intervals fulfill the inequality. We then use this information to graph the solution on a number line, shading the intervals where the inequality holds true, and marking endpoints with open or closed circles according to whether the interval includes the endpoint or not.