Solving and Graphing Quadratic Inequalities

Solve the inequality $x^2+5x\ge -6$.

Solution

Step 1: Bringing –6 to the left-hand side of the inequality, we obtain

$x^2+5x+6\ge 0$

Factorizing this quadratic inequality yields

$(x+2)(x+3)\ge 0$

Step 2: We now need to find the roots of inequality. Our first instinct here may be to use the Zero Product Property. However, note that the Zero Product Property is used for equations, not inequalities. Instead, we need to solve for the x-intercepts by switching the inequality to an equation and then adjusting the inequality sign for the situation at hand in terms of the found x-intercepts. This is shown below.

$$\begin{gathered} (x+2)(x+3)=0 \\ \Rightarrow x+2=0 \\ \Rightarrow x=-2 \\ and \\ x+3=0 \\ \Rightarrow x=-3 \end{gathered}$$

Step 3: From the table, we find that this inequality obeys Case 2 with a > 0. Since y is positive, we must choose the values of x for which the curve is above the x-axis.

Step 4: Now, writing the solution in interval notation, we obtain

$x\le -2andx\ge -3$

The graph is shown below.

Example 1, Aishah Amri - StudySmarter Originals

Solve the inequality $10x^2<19x-6$.

Solution

Step 1: Bringing 19x and –6 to the left-hand side of the inequality, we obtain

$10x^2-19x+6<0$

Factorizing this quadratic inequality yields

$(5x-2)(2x-3)<0$

Step 2: As with the previous example, we shall treat our inequality above as an equation to determine its roots as below.

$$\begin{gathered} (5x-2)(2x-3)=0 \\ \Rightarrow 5x-2=0 \\ \Rightarrow x=\frac{2}{5} \\ and \\ 2x-3=0 \\ \Rightarrow x=\frac{3}{2} \end{gathered}$$

Step 3: Referring again to our table, we see that this inequality obeys Case 1 with a > 0. Since y is negative, we must choose the values of x for which the curve is below the x-axis.

Step 4: Thus, writing the solution in its corresponding interval notation form, we have

$\frac{2}{5}<x<\frac{3}{2}$

Example 2, Aishah Amri - StudySmarter Originals

Graph the quadratic inequality $y>-x^2+x+6$.

Solution

Step 1: We begin by graphing the polynomial $y=-x^2+x+6$

The coefficient of x² is negative, so the curve opens down. Factorizing this expression, we obtain

$$\begin{gathered} y=-(x^2-x-6) \\ \Rightarrow y=-(x+2)(x-3) \end{gathered}$$

Equating this to zero, we have roots at $x=-2andx=3$.

Since the inequality is >, our curve must be a dashed line. The graph is shown below.

Example 3 (1), Aishah Amri - StudySmarter Originals

Step 2: Let us now test a point inside the parabola, say (1, 2). Plugging x = 1 and y = 2 into our quadratic inequality, we find that

$$\begin{gathered} 2>-{\left(1\right)}^2+\left(1\right)+6 \\ \Rightarrow 2>6 \end{gathered}$$

Step 3: 2 is not more than 6, so the inequality fails. Thus, (1, 2) is not a solution to the inequality and so we must shade the region outside the parabola as shown below.

Example 3 (2), Aishah Amri - StudySmarter Originals

Now if we consider the quadratic inequality in one variable, we obtain

$-x^2+x+6<0$

Hence, y is negative and we must choose the values of x for which the curve is below the x-axis. The final graph is shown below.

Example 3 (3), Aishah Amri - StudySmarter Originals

Graph the quadratic inequality $y\le 5x^2-10x+1$.

Solution

Step 1: As before, we shall attempt to first graph the polynomial $y=5x^2-10x+1$.

The coefficient of x² is positive, so the curve opens up. Notice that we cannot factorize this expression using standard factoring methods. Hence, we shall apply the Quadratic Formula to determine the roots.

Given that

$a=5,b=-10,c=1$

we evaluate

$$\begin{gathered} x=\frac{-b\pm \sqrt{b^2-4ac}}{2a} \\ \Rightarrow x=\frac{-(-10)\pm \sqrt{{(-10)}^2-4\left(5\right)\left(1\right)}}{2\left(5\right)} \\ \Rightarrow x=\frac{10\pm \sqrt{80}}{10} \\ \Rightarrow x=\frac{10\pm 4\sqrt{5}}{10} \\ \Rightarrow x=1\pm \frac{2\sqrt{5}}{5} \end{gathered}$$

Thus, we obtain two irrational roots

$$\begin{gathered} x=1-\frac{2\sqrt{5}}{5}\approx 0.11(correctto2decimalplaces) \\ x=1+\frac{2\sqrt{5}}{5}\approx 1.89(correctto2decimalplaces) \end{gathered}$$

Since the inequality is ≤, our curve must be a solid line. The graph is shown below.

Example 4 (1), Aishah Amri - StudySmarter Originals

Step 2: Let us now check a point inside the parabola, say (1, –1). Plugging x = 1 and y = –1 into out inequality, we find that

$$\begin{gathered} -1\le 5{\left(1\right)}^2-10\left(1\right)+1 \\ \Rightarrow -1\le -4 \end{gathered}$$

Step 3: –1 is not less than or equal to –4, so the inequality fails. Thus, (1, –1) is not a solution to the inequality and so we must shade the region outside the parabola as below.

Example 4 (2), Aishah Amri - StudySmarter Originals

Now if we consider the quadratic inequality in one variable, we obtain

$5x^2-10x+6\ge 0$

Hence, y is positive and we must choose the values of x for which the curve is above the x-axis. The final graph is shown below.

Example 4 (3), Aishah Amri - StudySmarter Originals