Quadratic Inequalities

GCSE Maths Algebra Quadratic Inequalities

Quadratic inequalities are mathematical expressions comparing the value of a quadratic function to zero, with the general form:

$ax^2 + bx + c < 0$

$ax^2 + bx + c > 0$

$ax^2 + bx + c \le 0$

$ax^2 + bx + c \ge 0$

Quadratic inequalities are similar to quadratic equations, but they involve inequality signs instead of equal signs. Solving quadratic inequalities requires finding the range of values for the variable that satisfies the inequality.

Solving Quadratic Inequalities

Solving Quadratic Inequalities by Factorisation

Factorisation is an algebraic method that can solve quadratic inequalities. To factorise quadratic inequalities, follow these steps:

Step 1: Write the quadratic inequality in the form: $ax^2 + bx + c > 0$ or $ax^2 + bx + c < 0$ .

Step 2: Factor the quadratic expression on the left side of the inequality.

Step 3: Identify the critical points (the x-values that make the inequality equal to 0) and use them to form intervals.

Step 4: Test a point within each interval to determine where the inequality holds true.

Let’s look at an example:

Solve the inequality $x^2 - 6x + 8 > 0$ by factorisation.

1. The inequality $x^2 - 6x + 8 > 0$ is already in the form $ax^2 + bx + c > 0$ .

2. Factor the quadratic expression: $(x - 2)(x - 4) > 0$ .

3. Identify the critical points:

The critical points are $x = 2$ and $x = 4$ because these are the values of x that make the factored expression $(x - 2)(x - 4)$ equal to 0. If you set each factor to 0, you’ll find the critical points:

$x - 2 = 0$ → $x = 2$

$x - 4 = 0$ → $x = 4$

Divide the number line into intervals using these points: $x < 2$ , $2 < x < 4$ , and $x > 4$ .

4. Test a point within each interval to determine where the inequality holds true:

a) $x < 2$ : Choose $x = 1$

$(1 - 2)(1 - 4) = (-1)(-3) = 3$ , which is greater than 0.

b) $2 < x < 4$ : Choose $x = 3$

$(3 - 2)(3 - 4) = (1)(-1) = -1$ , which is not greater than 0.

c) $x > 4$ : Choose $x = 5$

$(5 - 2)(5 - 4) = (3)(1) = 3$ , which is greater than 0.

Based on the results of the test points, the solution set for the inequality $x^2 - 6x + 8 > 0$ is $x < 2$ or $x > 4$ .

Solving Quadratic Inequalities Using Quadratic Formula

Using the quadratic formula is another algebraic method that can solve quadratic inequalities. To solve inequalities using the quadratic formula, follow these steps:

Step 1: Write the quadratic inequality in standard form: $ax^2 + bx + c > 0$ or $ax^2 + bx + c < 0$ .

Step 2: Solve the corresponding quadratic equation $ax^2 + bx + c = 0$ using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

These solutions will be the critical points that form the intervals

Step 3: Identify the intervals where the inequality holds true by testing a point within each interval.

Let’s look at an example:

Solve the inequality $2x^2 - 5x - 3 < 0$ using the quadratic formula.

1. The inequality is already in standard form: $2x^2 - 5x - 3 < 0$ .

2. Solve the quadratic equation $2x^2 - 5x - 3 = 0$ using the quadratic formula:

$x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(-3)}}{2(2)}$

$x = \frac{5 \pm \sqrt{25 + 24}}{4}$

$x = \frac{5 \pm \sqrt{49}}{4}$

$x = \frac{5 \pm 7}{4}$

The solutions are $x = \frac{-1}{2}$ and $x = 3$ . These are the critical points that divide the number line into intervals: $x < \frac{-1}{2}$ , $\frac{-1}{2} < x < 3$ , and $x > 3$ .

3. Test a point within each interval to determine where the inequality holds true:

a) $x < \frac{-1}{2}$ : Choose $x = -1$ :

$2(-1)^2 - 5(-1) - 3 = 10$ , which is not less than 0.

b) $\frac{-1}{2} < x < 3$ : Choose x = 1:

$2(1)^2 - 5(1) - 3 = -6$ , which is less than 0.

c) $x > 3$ : Choose $x = 4$ :

$2(4)^2 - 5(4) - 3 = 21$ , which is not less than 0.

Based on the results of the test points, the solution set for the inequality $2x^2 - 5x - 3 < 0$ is $\frac{-1}{2} < x < 3$ .

Graphical Representation of Quadratic Inequalities

A quadratic equation of the form $ax^2 + bx + c = 0$ has at most two solutions. These solutions are the x-intercepts (the points at which the parabola intersects the x-axis) of the corresponding quadratic function $y = ax^2 + bx + c$ . The quadratic equation represents the specific x-values where the function equals zero.

On the other hand, a quadratic inequality such as $ax^2 + bx + c > 0$ or $ax^2 + bx + c < 0$ has a range of x-values as its solution, rather than just two distinct points. The solution set for a quadratic inequality consists of intervals where the parabola lies above or below the x-axis, depending on the inequality sign.

When solving quadratic inequalities graphically, the graph should show the following:

The parabola $y = ax^2 + bx + c$ representing the quadratic function.

The x-intercepts (if any) of the parabola, which are the solutions of the quadratic equation $ax^2 + bx + c = 0$ .

The intervals where the parabola is above or below the x-axis, based on the inequality sign. For example, if the inequality is ax^2 + bx + c > 0, the solution set consists of the intervals where the parabola is above the x-axis.

Solving Quadratic Inequalities Graphically

To solve a quadratic inequality graphically, you can follow these steps:

Step 1: Rewrite the inequality in the form $ax^2 + bx + c > 0$ or $ax^2 + bx + c < 0.$

Step 2: Draw the corresponding quadratic function $y = ax^2 + bx + c$ on a graph.

Step 3: Identify the x-intercepts (if any) of the parabola, which are the points where the parabola intersects the x-axis. These x-intercepts correspond to the critical points where the quadratic expression is equal to zero.

Step 4: Determine the intervals where the parabola is above or below the x-axis, based on the inequality sign.

Step 5: Combine the intervals where the inequality is true to form the solution set.

Now let’s look at two contrasting examples:

Greater than zero

If we solve the quadratic inequality $x^2 - 6x + 5 > 0$ graphically:

1. We draw the quadratic function $y = x^2 - 6x + 5$ , which forms a parabola.

2. The x-intercepts are $x = 1$ and $x = 5$ , which are the solutions of the quadratic equation $x^2 - 6x + 5 = 0$ . By factoring, we get $(x - 1)(x - 5) = 0$ .

3. The solution set for the inequality $x^2 - 6x + 5 > 0$ consists of the intervals where the parabola is above the x-axis. In this case, the graph shows that the parabola is above the x-axis for x < 1 and x > 5. These intervals represent the solution set for the quadratic inequality.

So, the solution set for the quadratic inequality $x^2 - 6x + 5 > 0$ is $x < 1$ or $x > 5$ .

Less than zero

If we solve the quadratic inequality $x^2 - 6x + 5 < 0$ graphically:

1. We draw the quadratic function $y = x^2 - 6x + 5$ , which forms a parabola.

2. The x-intercepts are $x = 1$ and $x = 5$ , which are the solutions of the quadratic equation $x^2 - 6x + 5 = 0$ . By factoring, we get $(x - 1)(x - 5) = 0$ .

3. The solution set for the inequality $x^2 - 6x + 5 < 0$ consists of the intervals where the parabola is below the x-axis. In this case, the graph shows that the parabola is below the x-axis for 1 < x < 5. This interval represents the solution set for the quadratic inequality.

So, the solution set for the quadratic inequality $x^2 - 6x + 5 < 0 is 1 < x < 5$ .

Greater than zero vs less than zero

Comparing the two examples:

For the inequality $x^2 - 6x + 5 > 0$ , the solution set is $x < 1$ or $x > 5$ . The graph shows the parabola above the x-axis in these intervals.

For the inequality $x^2 - 6x + 5 < 0$ , the solution set is $1 < x < 5$ . The graph shows the parabola below the x-axis in this interval.

Examples

Example 1:

Solve the inequality $x^2 - 4x + 3 > 0$ .

1. First, factor the quadratic expression: $(x - 1)(x - 3) > 0$ .

2. Identify the critical points: $x = 1$ and $x = 3$ .

3. Test the intervals between the critical points to find where the inequality holds true. In this case, test $x < 1$ , $1 < x < 3$ , and $x > 3$ .

4. The inequality is true when $x < 1$ and $x > 3$ .

The solution set for this inequality is $x < 1$ or $x > 3$ .

Example 2:

Solve the inequality $x^2 - 6x + 8 \le 0$ .

1. Factor the quadratic expression: $(x - 2)(x - 4) \leq 0$ .

2. Identify the critical points: $x = 2$ and $x = 4$ .

3. Test the intervals between the critical points: $x < 2, 2 \leq x \leq 4$ , and $x > 4$ .

4. The inequality is true when $2 \leq x \leq 4$ .

The solution set for this inequality is $2 \leq x \leq 4$ .

Example 3:

A rectangular garden has a width of x meters and a length of 2x meters. The garden’s area must be less than or equal to 60 square meters. Determine the possible values of x.

1. Write the inequality: $x(2x) \leq 60$ .

2. Simplify the inequality: $2x^2 \leq 60$ .

3. Divide both sides by 2: $x^2 \leq 30$ .

4. To find the possible values of x, consider the square root of both sides: $-\sqrt{30} \leq x \leq \sqrt{30}$ .

The solution set for this inequality is $-\sqrt{30} \leq x \leq \sqrt{30}$ . However, since width cannot be negative, the practical solution set for this problem is $0 < x \leq \sqrt{30}$ .

Example 4:

Solve the quadratic inequality $x^2 - 4x + 3 > 0$ graphically.

1. The inequality is already in standard form: $x^2 - 4x + 3 > 0$

2. Draw the corresponding quadratic function $y = x^2 - 4x + 3$ on a graph, which will form a parabola.

3. Identify the x-intercepts of the parabola. In this case, the parabola intersects the x-axis at $x = 1$ and $x = 3$ . These are the critical points.

4. Determine the intervals where the parabola is above the x-axis, as the inequality is greater than zero $(x^2 - 4x + 3 > 0)$ . Based on the graph, the parabola is above the x-axis for $x < 1$ and $x > 3$ .

5. The solution set for the inequality $x^2 - 4x + 3 > 0$ is $x < 1$ or $x > 3$ .

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Last updated: April 15, 2022

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• Performance of a contract: The provision of personal data is necessary for the performance of an agreement with you or for any pre-contractual obligations thereof.

• Legal obligations: Processing personal data is necessary for compliance with a legal obligation to which the company is subject.

• Vital interests: Processing personal data is necessary in order to protect your vital interests or of another natural person.

• Public interests: Processing personal data is related to a task that is carried out in the public interest or in the exercise of official authority vested in the Company.

• Legitimate interests: Processing personal data is necessary for the purposes of the legitimate interests pursued by the Company.

In any case, we will gladly help to clarify the specific legal basis that applies to the processing, and in particular whether the provision of personal data is a statutory (required by law) or a contractual requirement, or a requirement necessary to enter into a contract.

Your Rights under the GDPR

We respect the confidentiality of your personal data and we guarantee that you can exercise your rights.

You have the right under this privacy policy, and by law if you are within the EU or the UK, to:

• Request access to your personal data. The right to access, update or delete the information we have on you. Whenever made possible, you can access, update or request deletion of your personal data directly within your account settings section. If you are unable to perform these actions yourself, please contact us to assist you. This also enables you to receive a copy of the personal data we hold about you.

• Request correction of the personal data that we hold about you. You have the right to have any incomplete or inaccurate information we hold about you corrected.

• Object to processing of your personal data. This right exists where we are relying on a legitimate interest as the legal basis for our processing and there is something about your particular situation, which makes you want to object to our processing of your personal data on this ground. You also have the right to object to us processing your personal data for direct marketing purposes.

• Request erasure of your personal data. You have the right to ask us to delete or remove personal data when there is no good reason for us to continue processing it.

• Request the transfer of your personal data. We will provide to you, or to a third-party you have chosen, your personal data in a structured, commonly used, machine-readable format. Please note that this right only applies to automated information which you initially provided consent for us to use or where we used the information to perform a contract with you.

• Withdraw your consent. You have the right to withdraw your consent on using your personal data. If you withdraw your consent, we may not be able to provide you with access to specific functionalities of our services.

Exercising of Your GDPR Data Protection Rights

You may exercise Your rights of access, rectification, cancellation and opposition by contacting us. Please note that we may ask you to verify your identity before responding to such requests. If you make a request, we will try our best to respond to you as soon as possible.

You have the right to complain to a Data Protection Authority about our collection and use of your personal data. For more information, if you are in the European Economic Area (EEA), please contact Your local data protection authority in the EEA.

Children’s Privacy

Our website may contain content appropriate for children under the age of 13. As a parent, you should know that by using our services, children under the age of 13 may participate in activities that involve the collection or use of personal information. We use reasonable efforts to ensure that before we collect any personal information from a child, the child’s parent receives notice of and consents to our personal information practices.

We may also limit how we collect, use, and store some of the information of users between 13 and 18 years old. In some cases, this means that we will be unable to provide certain functionalities of our services to these users. If we need to rely on consent as a legal basis for processing your information and your country requires consent from a parent, We may require the parent’s consent before we collect and use that information.

We may ask a user to verify their date of birth before collecting any personal information from them. If the user is under the age of 13, the service will either be blocked or redirected to a parental consent process.

Information Collected from Children Under the Age of 13

We may collect and store persistent identifiers, such as cookies or IP addresses from children, without parental consent for the purpose of supporting the internal operation of the our services.

We may collect and store other personal information about children if this information is submitted by a child with prior parental consent or by the parent or guardian of the child.

We may collect and store the following types of personal information about a child when submitted by a child with prior parental consent or by the parent or guardian of the child:

First and/or last name
Date of birth
Gender
Grade level/year group
Email address
Telephone number
Parent’s or guardian’s name
Parent’s or guardian’s email address
For further details on the information we might collect, you can refer to the “Types of Data Collected” section of this privacy policy. We follow our standard privacy policy for the disclosure of personal information collected from and about children.

Parental Access

A parent who has already given the Company permission to collect and use their child’s personal information can, at any time:

Review, correct or delete the child’s personal information
Discontinue further collection or use of the child’s personal information
To make such a request, you can write to us using the contact information provided in this privacy policy.

Links to Other Websites

Our Service may contain links to other websites that are not operated by us. If you click on a third party link, you will be directed to that third party’s site. We strongly advise you to review the privacy policy of every site you visit.

We have no control over and assume no responsibility for the content, privacy policies or practices of any third party sites or services.

Policies Unique to Our Tutoring Services

The following applies to students that have registered for our tutoring services and tutors and employees representing Shalom Education Ltd:

Your profile includes information such as your username and password, purchases or orders made by you, your interests, preferences, and feedback and survey responses.

Booking history including booking requests, subjects booked, set-up test while using our digital resources, any online meeting and recordings of your bookings

Electronic recordings from CCTV, tutoring session, video conference calls, communication history with other users, your interaction with Shalom Education Tutors, records of referral codes shared, redeemed and any promotions that have been applied to your account. All photo’s, videos and conference recordings may be used to protect learner’s, Tutors and employees and may be used as evidence to support any complaint.

Changes to this Privacy Policy

We may update Our Privacy Policy from time to time. We will notify You of any changes by posting the new Privacy Policy on this page.

We may let You know via email and/or a prominent notice on our website prior to the change becoming effective and update the “Last updated” date at the top of this Privacy Policy.

You are advised to review this Privacy Policy periodically for any changes. Changes to this Privacy Policy are effective when they are posted on this page.

Contact Us

If you have any questions about this Privacy Policy, You can contact us:

By email: info@shalom-education.com
By visiting this page on our website: https://www.shalom-education.com/contact
By phone number: 01206657616

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Cookie	Duration	Description
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