Chapter P.7, Problem 31E

Solution

a.

An example of a quadratic inequality with a solution of all real numbers.

  $x^2\ge 0$

Given:

All real numbers are solution of a quadratic inequality.

Concept Used:

• If there is the strict inequality symbol $<\text{ or}>$ , then use a parenthesis $\left(\right)$ on the interval
• If there is the inequality symbol $\le \text{ or}\ge$ , then use a bracket $\left[\right]$ on the interval
• To graph the endpoint with strict inequality $<\text{ or}>$, use the open parenthesis or a hollow circle at that endpoint.
• To graph the endpoint with inequality symbol $\le \text{ or}\ge$, use the bracket or a solid circle at that endpoint.
• Multiplying or dividing an inequality by a negative number reverse the inequality sign.

Calculation:

In order to find a quadratic inequality that has all real numbers as solution, it must include all positive and negative real numbers including 0.

Thus, required inequality is $x^2\ge 0$ .

b.

An example of a quadratic inequality with no solution.

  $x^2<0$

Given:

The quadratic inequality has no solution.

Concept Used:

• If there is the strict inequality symbol $<\text{ or}>$ , then use a parenthesis $\left(\right)$ on the interval
• If there is the inequality symbol $\le \text{ or}\ge$ , then use a bracket $\left[\right]$ on the interval
• To graph the endpoint with strict inequality $<\text{ or}>$, use the open parenthesis or a hollow circle at that endpoint.
• To graph the endpoint with inequality symbol $\le \text{ or}\ge$, use the bracket or a solid circle at that endpoint.
• Multiplying or dividing an inequality by a negative number reverse the inequality sign.

Calculation:

In order to find a quadratic inequality that has no solution, use the concept that the square of any real number is always positive.

Thus, required inequality is $x^2<0$ .

c.

An example of a quadratic inequality with exactly one solution.

  $x^2\le 0$

Given:

The quadratic inequality exactly one solution.

Concept Used:

• If there is the strict inequality symbol $<\text{ or}>$ , then use a parenthesis $\left(\right)$ on the interval
• If there is the inequality symbol $\le \text{ or}\ge$ , then use a bracket $\left[\right]$ on the interval
• To graph the endpoint with strict inequality $<\text{ or}>$, use the open parenthesis or a hollow circle at that endpoint.
• To graph the endpoint with inequality symbol $\le \text{ or}\ge$, use the bracket or a solid circle at that endpoint.
• Multiplying or dividing an inequality by a negative number reverse the inequality sign.

Calculation:

In order to find a quadratic inequality that has no solution, use the concept that the square of any real number is always positive, so 0 could be the only one solution.

Thus, required inequality is $x^2\le 0$ .

d.

An example of a quadratic inequality with a solution $\left[-2,5\right]$ .

  $\left(x+2\right)\left(x-5\right)\le 0$

Given:

The solution of quadratic inequality is $\left[-2,5\right]$ .

Concept Used:

• If there is the strict inequality symbol $<\text{ or}>$ , then use a parenthesis $\left(\right)$ on the interval
• If there is the inequality symbol $\le \text{ or}\ge$ , then use a bracket $\left[\right]$ on the interval
• To graph the endpoint with strict inequality $<\text{ or}>$, use the open parenthesis or a hollow circle at that endpoint.
• To graph the endpoint with inequality symbol $\le \text{ or}\ge$, use the bracket or a solid circle at that endpoint.
• Multiplying or dividing an inequality by a negative number reverse the inequality sign.

Calculation:

In order to find a quadratic inequality that has the solution set $\left[-2,5\right]$ , it must include all real numbers that are greater than or equal to -2 and less than or equal to 5.

Thus, required inequality is $\left(x+2\right)\left(x-5\right)\le 0$ .

e.

An example of a quadratic inequality with a solution $\left(-\infty ,-1\right)\cup \left(4,\infty \right)$ .

  $\left(x+1\right)\left(x-4\right)>0$

Given:

The quadratic inequality has solution $\left(-\infty ,-1\right)\cup \left(4,\infty \right)$ .

Concept Used:

• If there is the strict inequality symbol $<\text{ or}>$ , then use a parenthesis $\left(\right)$ on the interval
• If there is the inequality symbol $\le \text{ or}\ge$ , then use a bracket $\left[\right]$ on the interval
• To graph the endpoint with strict inequality $<\text{ or}>$, use the open parenthesis or a hollow circle at that endpoint.
• To graph the endpoint with inequality symbol $\le \text{ or}\ge$, use the bracket or a solid circle at that endpoint.
• Multiplying or dividing an inequality by a negative number reverse the inequality sign.

Calculation:

In order to find a quadratic inequality that has the solution set $\left(-\infty ,-1\right)\cup \left(4,\infty \right)$ , it must include all real numbers that are less than −1 and greater than 4.

Thus, required inequality is $\left(x+1\right)\left(x-4\right)>0$ .

f.

An example of a quadratic inequality with a solution $\left(-\infty ,0\right]\cup \left[4,\infty \right)$ .

  $x\left(x-4\right)\ge 0$

Given:

The quadratic inequality has solution $\left(-\infty ,0\right]\cup \left[4,\infty \right)$ .

Concept Used:

• If there is the strict inequality symbol $<\text{ or}>$ , then use a parenthesis $\left(\right)$ on the interval
• If there is the inequality symbol $\le \text{ or}\ge$ , then use a bracket $\left[\right]$ on the interval
• To graph the endpoint with strict inequality $<\text{ or}>$, use the open parenthesis or a hollow circle at that endpoint.
• To graph the endpoint with inequality symbol $\le \text{ or}\ge$, use the bracket or a solid circle at that endpoint.
• Multiplying or dividing an inequality by a negative number reverse the inequality sign.

Calculation:

In order to find a quadratic inequality that has the solution set $\left(-\infty ,0\right]\cup \left[4,\infty \right)$ , it must include all real numbers that are less than or equal to 0 and greater than or equal to 4.

Thus, required inequality is $x\left(x-4\right)\ge 0$ .