Question

Solve the inequality by graphing. $$x^2+13 x+42<0$$

Short answer

The solution for the inequality \(x^2 + 13x + 42<0\) is -7<x<-6

Step-by-step solution

Factor the quadratic equation

Factor the quadratic equation \(x^2 + 13x + 42\). This can be done by finding two numbers that sum up to 13 and multiply to 42. The numbers 6 and 7 satisfy these criteria. So, the expression can be factored as \((x+6)(x+7)\).

Find the roots

The roots of the equation are the x-values that make the equation equal to zero. They can be found by setting the factored equation \((x+6)(x+7)=0\) equal to zero and solving for x. This will give the roots x=-6 and x=-7.

Plot the roots on the number line and perform test points

Plot points -7 and -6 on the number line. Segment the number line into three parts using these plots and choose test points from each part, for instance -8 from (-∞,-7), -6.5 from (-7,-6), and -5 from (-6,∞). Plug these values into the original equation and check whether the inequality holds.

Interpret the results

The inequality holds true for values between -7 and -6 and for these values the expression is less than zero.

Key concepts

Factoring Quadratic Equations

Factoring a quadratic equation is often the first step in solving it. Quadratic equations are typically in the form of \(ax^2 + bx + c\). The goal is to express it as a product of two binomials. In other words, we look for two numbers that add up to the coefficient of the middle term \(b\) and multiply to \(c\). For example, in the equation \(x^2 + 13x + 42\), we need two numbers that sum to 13 and multiply to 42. The numbers 6 and 7 fit perfectly, allowing us to factor the quadratic into \((x+6)(x+7)\). This step simplifies the equation, making it easier to solve or further analyze.

Finding Roots of Equations

Once the quadratic equation is factored, finding the roots becomes straightforward. Roots are the values of \(x\) that make the equation equal to zero. For the equation \((x+6)(x+7)=0\), you set each binomial to zero separately:

• \(x+6=0\)
• \(x+7=0\)

Solving these equations gives us the roots \(x=-6\) and \(x=-7\). These roots are the solutions where the graph of the quadratic equation intersects the x-axis. Understanding this concept is crucial, especially when dealing with inequalities, as it helps determine the intervals to test for the solution.

Graphing Solutions

Graphing a quadratic inequality involves understanding the related quadratic equation's graph. The graph of \(y=x^2 + 13x + 42\) is a parabola. For inequalities like \(x^2 + 13x + 42 < 0\), we are interested in where this parabola is below the x-axis. Specifically, we want the part of the graph that sits underneath it. On a coordinate plane, we plot the roots \(x=-6\) and \(x=-7\), which divide the x-axis into distinct sections. The parabola opens upwards, so it dips below the x-axis between these two roots. Therefore, values of \(x\) between \(-7\) and \(-6\) satisfy the inequality \(x^2 + 13x + 42 < 0\). This visual representation helps in understanding which part of the number line meets the inequality condition.

Number Line Analysis

A number line analysis is a valuable tool when solving quadratic inequalities. Once we have the roots of the equation, placing them on a number line helps identify intervals to test. For the problem \(x^2 + 13x + 42 < 0\), we plot the roots \(-6\) and \(-7\) on the line. These points divide the line into three intervals:

• \((-\infty, -7)\)
• \((-7, -6)\)
• \((-6, \infty)\)

We select test points from each interval, such as \(-8\), \(-6.5\), and \(-5\), and substitute them back into the original inequality. Only the values in \((-7, -6)\) will satisfy \(x^2 + 13x + 42 < 0\), as the expression is negative here. This method not only helps verify solutions graphically but also reinforces understanding of inequality concepts.