Equations and Inequalities

Q: How do you solve equations and inequalities?

Equations and inequalities that contain single variables get their variables isolated. After the arithmetic operations have been performed, the value equated to the variable is the solution. Equations and inequalities that contain two variables are solved simultaneously.

Q: How to graph linear equations and inequalities?

Rewrite them in slope-intercept form, then use the zero product property to find the values of the variables which will give you two points on the coordinate plane. There you can plot what values you have.

Q: What are equations and inequalities examples?

Examples of equations are; 2x+1=5 4 - 2x = 12 Examples of inequalities are; 1+3<5x 21 > 6x -1

Q: What is equation and inequality?

An equation is a formula that expresses the equality of two expressions by connecting them with the equals sign (=). Inequalities, similar to equations, are also mathematical statements that rather possess a greater than (>), greater than or equal to (≥), less than (<), or less than or equal to (≤), between expressions in place of the equal sign in equations.

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Did you know that there are mathematical concepts that could be used to compare the values of quantities to see whether they are equal, or whether some are greater than or less than others?

In this article, we are going to discuss equations and inequalities.

Equations and Inequalities definition

Equations definition

An equation expresses the equality of two expressions through the equals sign (=).

The above definition literally states that when an expression is equated to another expression, both expressions are equal.

An example of an equation looks like $3 x + 2 = 2$ .

Variables in equations are placeholders and they represent unknown quantities that require solving. For this reason, real-life problems are modeled into equations to be solved.

If I spend $100 on a pair of shoes and a shirt, and I only know the cost of the shirt to be $40. How will I know the cost of the pair of shoes?

Modeling this mathematically will have us equate everything we purchased to $100 since that is the total of what was spent.

We represent the unknown quantity with a variable, $x$ , and thus we have

$$ 40 + x = $ 100$ .

We can then tell from the equation that the cost of the pair of shoes was $60 since 40+60=100.

Inequalities definition

Inequalities are mathematical statements that rather possess a greater than (>), greater than or equal to (≥), less than (<), or less than or equal to (≤), between expressions in place of the equal sign in equations.

An example of inequality looks like $4 - 3 x < 16$ .

Inequalities are used to model and represent quantities that are not particularly equal to each other.

Representing ranges is perfectly done by inequalities.

We use the minimum wage as an example.

The government declares that the acceptable minimum wage in the country is now $10 per hour.

This means mathematically is that what companies are allowed to pay their employees is now $x \geq 10$ , where $x$ is the hourly rate.

Equations and inequalities examples

The table below displays examples of equations and inequalities.

Equations	Inequalities
$2 x - 5 = 12$	$x + 3 > 2$
$24 y + 2 x = y$	$5 \geq x - 1$
$2^{x + 1} = 16$	$(x + 4) (x - 2) (x - 7) < 0$
$3 (1 + x) = 4 x - 3$	$x^{2} - 5 x + 6 \leq 0$

Equations and inequalities differences

Comparatively, there are quite some differences between equations and inequalities. These are outlined in the table below.

Equations	Inequalities
These express how equal different quantities are.	These express how unequal the different quantities are.
The symbol used is the equal sigh (=)	Symbols used are greater than (>), greater than or equal to (≥), less than (<), and less than or equal to (≤).
These have a fixed and single solution.	Solutions are usually included in an interval, particularly many times.
The total number of roots for equations is definite.	The total number of roots for inequalities is infinite.

Polynomial equations and inequalities

We now introduce the definitions of polynomial equations and inequalities.

A polynomial equation is an equation involving polynomials.

We recall that a polynomial is of the form

$a_{n} x^{n} + . . . + a_{1} x + a_{0} = 0$

$a_{n}, . . ., a_{1}, a_{0}$ are the coefficients
$x$ is the variable
$n$ is the highest exponent, positive integer.

Examples of polynomial equations are,

Example	Type	Description
$4 x + 4 - 12 = 0$	Linear Equations	Equations of degree 1
$6 x^{2} + 11 x - 35 = 0$	Quadratic Equations	Equations of degree 2
$2 x^{3} + 3 x^{2} + 14 x + 2 = 0$	Cubic Equations	Equations of degree 3

A polynomial inequality is an inequality involving polynomials.

$x^{2} + 3 x - 1 < 0 x^{2} + 2 \geq 0$

The difference between polynomial equations and inequalities are the signs that are associated with them.

Polynomial equations and inequalities examples

Here, we are going to take examples of how real-life situations are modelled into equations to accelerate problem-solving.

Summing two consecutive numbers gives us 61. Find these numbers

Solution

Mathematically, consecutive numbers are ones that follow each other directly. This means when we sum such numbers, we should have 61.

Let us represent the first number with the variable $x$ . Since the second number is consecutive to the first, that can be represented by $x + 1$ .

All we need to do now is to sum these two numbers and equate them to 61 as the problem states,

$x + x + 1 = 61$

$2 x + 1 = 61$

We now have the equation for the problem. We can go ahead to verify this by solving for x to see what these consecutive numbers are.

$2 x + 1 = 61$

Isolate $2 x$ by subtracting 1 from each side of the equation,

$2 x + 1 - 1 = 61 - 1$

$2 x = 60$

Divide both sides by 2 to get,

$\frac{2 x}{2} = \frac{60}{2}$

$x = 30$

This means that the first number is 30.

Since the second number was represented as $x + 1$ , we will substitute 30 into this expression.

$30 + 1 = 31$

We now have the second number to be 31.

Now let us substitute these numbers into the equation we modelled to get,

$x + x + 1 = 61$

$30 + 30 + 1 = 61$

Mike ordered a shirt and a pair of shoes for $100 and the shirt costs $45, how much did he spend on the pair of shoes?

Solution

Two items were purchased here, which sums up to $100.

The value for one was given, and the other wasn't. This means we can represent the item whose value wasn't given by a variable x.

$S h i r t \cos t = $ 45$

$P a i r o f s h o e s \cos t = x$

$T o t a l \cos t = $ 100$

We had the total by summing up the cost of the two items,

$$ 45 + x = $ 100$

Here, we have the problem modelled into an equation.

Isolating x, we get

$x = 100 - 45 = 55$

Thus, the price of the pair of shoes is 55 dollars.

We are also going to take examples of how real-life situations are modelled into inequalities to accelerate problem-solving.

If Adam scored 18 points in a game of tennis he played with Ben, and Ben scored at least 3 points more than Adam did, how many points did Ben score?

Solution

In dealing with inequalities, one thing we need to be extremely vigilant about is the sign involved. We see that Ben has a higher score than Adam, hence the sign will have to depict that.

Moreover, we have the words "at least" to take note of. This means if we are expressing that Ben's score is greater, it'll have to be greater than or equal to Adam's score.

Let Ben's score be x. To get Ben's score, we need to sum at least 3 and 18, and thus this can be modeled as

$x - 3 \geq 18$

13 students at least didn't make it to school on the 1st of July.

Solution

Again, at least depicts the greater or equal sign. And this problem means that the number of students who did not attend school on the said day is 13 or more. This is expressed as,

$x \geq 13$

Linear equations and inequalities in two variables

There are situations where equations and inequalities come in two variables to be solved. We are going to explore how they are being solved in this section.

Two-variable linear equations

Two-variable linear equations, as the name says are linear equations that have two variables. They are of the form,

$a x + b y = c$

where x and y are variables and a, b, and c are integers.

$2 x + 3 y = 8$ and $- x + 2 y = - 19$ are linear equations in two variables.

If Mike bought a pair of shoes and a shirt for $100, and Steve ordered the same pair of shoes and two shirts for $145, how much does the pair of shoes cost?

Solution

The extra shirt must have been $45 then. Which would make the pair of shoes in both equations $55. However, this can be solved mathematically if we model the equation right.

We can assign each unknown quantity to a variable.

Let $x = p a i r o f s h o e s p r i c e$

$y = s h i r t p r i c e$

Both the pair of shoes and the shirt costs $100. This means for a first equation, we will sum those variables and equate them to $100.

$x + y = 100$

With the second equation, the shirts are two. However, the total price for this purchase is $145 instead, thus we have

$x + 2 y = 145$

We now have both equations modelled. This is written as,

$\{\begin{cases} x + y = 100 \\ x + 2 y = 145 \end{cases}$

Equations and inequalities, Two variable equations, StudySmarter Mike and Steve's purchases-StudySmarter Originals

Two-variable linear inequalities

Two-variable linear inequalities are the linear inequalities that have two variables.

The only difference between them and two-variable linear equations are the signs involved. They are expressed in the form,

$a x + b y \leq c, a x + b y < c, a x + b y > c, o r a x + b y \geq c$

where, x and y are variables and a, b, and c are integers.

The signs involved in inequalities differ depending on the problem. We are going to take an example of a real-life situation and see how this helps solve them.

In order to get admitted in Grade 12, students must have a total score greater than 10.

This can be modelized as x>10 with x is the total score.

Quadratic functions equations and inequalities

The quadratic equations and inequalities are those that have their variables in the second degree. We are going to look at how they work in both equations and inequalities in this section.

Quadratic equations

Quadratic equations are polynomial equations of second degree. When graphed, they appear in a parabola shape. This makes them the ideal mathematical concept for calculating parabolic movements in real life such as the movements of rockets.

They are in standard form as $f (x) = a x^{2} + b x + c$ , where a, b, and c are real numbers with a ≠ 0.

However, this is not the only way quadratics are expressed. The forms in which quadratics can be expressed are,

The standard form: $f (x) = a x^{2} + b x + c, w h e r e a \neq 0$
The vertex form: $f (x) = a {(x - h)}^{2} + k, w h e r e a \neq 0 a n d (h, k)$ is the vertex of the parabola.
The intercept form: $f (x) = a (x - p) (x - q)$ where a ≠ 0 and (p, 0) and (q, 0) are the x-intercepts of the parabola.

In this article, we will only discuss how to model these equations. You can refer to the article on Solving Quadratic equations to learn more about how to solve such equations!

Which of the following equations is quadratic?

$2 x + 1 = 3$
$x^{2} = 4$
$\frac{1}{2} x + 4 y = 0$
$23 x^{3} + 4 y - 8 = 12$

Solution

The only quadratic equation here is

$x^{2} = 4$

since the highest degree of the polynomial here is 2.

Quadratic inequalities

Quadratic inequalities are second-degree polynomials possessing a greater than (>), greater than or equal to (≥), less than (<), or less than or equal to (≤), between expressions in place of the equal sign in equations.

Which of the following inequalities is quadratic?

$4 x^{3} - 18 \geq 3$
$23 x^{2} + 5 x < 0$
$x + y > 45$
$9 x < 8$

Solution

The only quadratic inequality here is

$23 x^{2} + 5 x < 0$ ,

since it has the standard form of a quadratic inequality.

Equations and Inequalities - Key takeaways

An equation expresses the equality of two expressions through the equals sign (=).
Inequalities are mathematical statements that rather possess a greater than (>), greater than or equal to (≥), less than (<), or less than or equal to (≤), between expressions in place of the equal sign in equations.
A polynomial equation is an equation involving polynomials.
Quadratic equations are the equations whose variables are in the second degree.
Quadratic inequalities are second-degree polynomials possessing a greater than (>), greater than or equal to (≥), less than (<), or less than or equal to (≤), between expressions.

Frequently Asked Questions about Equations and Inequalities

Equations and inequalities that contain single variables get their variables isolated. After the arithmetic operations have been performed, the value equated to the variable is the solution.

Equations and inequalities that contain two variables are solved simultaneously.

Rewrite them in slope-intercept form, then use the zero product property to find the values of the variables which will give you two points on the coordinate plane. There you can plot what values you have.

Examples of equations are;

2x+1=5

4 - 2x = 12

Examples of inequalities are;

1+3<5x

21 > 6x -1

In both equations and inequalities, what do you do to one side, has to be done on the other side

However, they are different because equations contain equal signs, whilst inequalities contain greater than (>), greater than or equal to (≥), less than (<), or less than or equal to (≤).

An equation is a formula that expresses the equality of two expressions by connecting them with the equals sign (=).

Inequalities, similar to equations, are also mathematical statements that rather possess a greater than (>), greater than or equal to (≥), less than (<), or less than or equal to (≤), between expressions in place of the equal sign in equations.

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Flashcards in Equations and Inequalities56 Start learning

Solve for x:

x²+2x+1=0

x=-1

Solve for x:

x²+2x+1=0

x=-1

Solve for x:

x²+2x+1=0

x=-1

What is a quadratic equation?

A quadratic equation is an equation that can be expressed in the form ax²+bx+c=0, where a≠0.

What is a quadratic equation?

A quadratic equation is an equation that can be expressed in the form ax²+bx+c=0, where a≠0.

What is a quadratic equation?

A quadratic equation is an equation that can be expressed in the form ax²+bx+c=0, where a≠0.

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