Inequalities Maths

Q: What is an inequality equation?

An inequality equation is an algebraic expression that instead of an equal symbol (=), contains the symbols less than (<), less than or equal to (≤), greater than (>), or greater than or equal to (≥).

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Inequalities are algebraic expressions that, instead of representing how both sides of an equation are equal to each other, represent how one term is less than, less than or equal, greater than, or greater than or equal than the other.

$x + 1 > 3$

This example is read as x plus 1 is greater than 3.

Notice that the arrowhead of the inequality symbol points to the smaller expression in an inequality.

Specifically, the symbols used in inequalities are:

symbol	Meaning
>	greater than
<	less than
$\geq$	greater than or equal
$\leq$	less than or equal

Properties of inequalities

The properties of inequalities are described in Table 1:

Table 1. Properties of inequalities

If a, b, and c are Real Numbers:

Property	Definition	Example
Addition	If $a > b$ , then $a + c > b + c$	$5 > 2$ , so $5 + 1 > 2 + 1$
Subtraction	If $a > b$ , then $a - c > b - c$	$6 > 3$ , so $6 - 2 > 3 - 2$
Multiplication	If $a > b$ and $c > 0$ , then $a \times c > b \times c$ If $a > b$ and $c < 0$ , then $a \times c < b \times c$	$4 > 2$ , and $3 > 0$ , so $4 \times 3 > 2 \times 3, 12 > 6$ $4 > 2$ , and $- 1 < 0$ , so $4 (- 1) < 2 (- 1), - 4 < - 2$
Division	If $a > b$ and $c > 0$ , then $\frac{a}{c} > \frac{b}{c}$ If $a > b$ and $c < 0$ , then $\frac{a}{c} < \frac{b}{c}$	$6 > 2$ , and $2 > 0$ , so $\frac{6}{2} > \frac{2}{2}$ , $3 > 1$ $4 > 2$ , and $- 1 < 0$ , so $\frac{4}{- 1} < \frac{2}{1}$ , $- 4 < - 2$
Transitive	If $a > b$ and $b > c$ , then $a > c$	$5 > 2$ and $2 > 1$ , so $5 > 1$
Comparison	If $a = b + c$ and $c > 0$ , then $a > b$	$5 = 2 + 3$ and $3 > 0$ , so $5 > 2$

What are the different types of inequalities?

The main types of inequalities that you can find are:

Linear inequalities

Linear inequalities are inequalities where the maximum exponent present in its variables is power 1.

$x + 2 < 7$

Quadratic inequalities

If the maximum exponent present in an inequality is power 2, it is called a quadratic inequality.

$x^{2} + x - 20 < 0$

Solving inequalities

To solve inequalities, you will have to follow different steps depending on whether they are linear or quadratic.

Solving linear inequalities

To solve linear inequalities, you can manipulate them to find a solution in the same way as an equation, keeping in mind the following extra rules:

The solution of an inequality is the set of all Real Numbers that make the inequality true. Therefore, any value of x that satisfies the inequality is a solution for x.
The symbols> (greater than) and <(less than) exclude the specific value as part of the solution. The symbols $\geq$ (greater than or equal) and $\leq$ (less than or equal) include the specific value as part of the solution instead of excluding it.
The solution of an inequality can be represented on the Number Line, using an empty circle to represent that the value of x is not part of the solution, and a closed circle if the value of x is part of the solution.
If you multiply or divide the inequality by a negative Number, then you need to reverse the symbol of the inequality. The best way to understand why you need to do this is to see an example.

You know that 4> 2, but if you multiply this inequality by -1

Then you get -4> -2 which is not true

For the inequality to remain true, you need to reverse the symbol, like this:

-4 <-2 ✔ which is true

This is because, in the case of negative numbers, the closer the Number is to zero, the larger it is.

You can see -4 and -2 represented on the Number Line as follows:

Inequalities Maths Numbers on number line StudySmarter Numbers on the number line, Marilú García De Taylor - StudySmarter Originals

If you have a fraction in an inequality where x is in the denominator (i.e. $\frac{4}{x} > 5$ ), you need to remember that x could be either positive or negative. Therefore, you can not multiply both sides of the inequality by x; multiply by $x^{2}$ instead so that the inequality continues to be true.

Examples of solving linear inequalities

1) x - 5> 8 isolate x and combine like terms

x> 8 + 5

x> 13

Using set notation, the solution is {x: x> 13}, which you can read as the set of values of x for which x is greater than 13.

2) 2x + 2 <16 isolate x and combine like terms

2x <16 -2

2x <14

$x < \frac{14}{2}$

x <7

Set notation: {x: x <7}

3) 5 - x <19

- x <19 - 5

- x <14 Remember to change the symbol, as you are dividing by -1

x> -14

Set notation: {x: x> -14}

4) If you need to find the set of values for which two inequalities are true together, you can use a number line to see the solution more CLEARLY.

The solution will be the values that satisfy both Equations at the same time. For example:

Inequalities Maths Solving linear inequalities number line StudySmarter Solving linear inequalities using the number line, Marilú García De Taylor - StudySmarter Originals

Set notation: {x: 4 <x <5}

If there is no overlap, then the inequalities are written separately.

Inequalities Maths Solving linear inequalities number line no overlap StudySmarter Solving linear inequalities using the number line - no overlap, Marilú García De Taylor - StudySmarter Originals

Set notation: {x: x <4} ∪ {x: x> 5}

Solving quadratic inequalities

To solve quadratic inequalities, you need to follow these steps :

1. Rearrange the terms to the left side of the inequality so that you have only zero on the other side.

You might need to expand brackets and combine like terms before solving a quadratic inequality.

2. Solve the quadratic equation to find the critical values. To do this, you can factorise, complete the square or use the quadratic formula.

3. Draw the graph of the quadratic function. The graph of a quadratic function ( $a x^{2} + b x + c > 0$ ) is a parabola that crosses the x-axis at the critical values. If the coefficient of $x^{2}$ (a) is negative, then the parabola will be upside down.

4. Use the graph to find the required set of values.

Examples of solving quadratic inequalities

Find the set of values of x for which $x^{2} + x - 6 > 0$

$x^{2} + x - 6 = 0$ factorise to find the critical values

(x - 2) (x + 3) = 0

The critical values are: x = 2 and x = -3

You can use a table to help you see where the graph will be positive or negative.

	x <-3	-3 <x <2	x> 2
(x - 2)	-	-	+
(x + 3)	-	+	+
(x - 2) (x + 3)	+	-	+

You can read the information on the table like this: If x <-3, (x - 2) is negative, (x + 3) is negative, and (x - 2) (x + 3) is positive, and the same for the other columns. The last row (x - 2) (x + 3) tells you where the graph will be positive or negative.

Now you can draw the graph:

Inequalities Maths Solving quadratic inequalities graph StudySmarter Solving quadratic inequalities graph, Marilú García De Taylor - StudySmarter Originals

The solution to $x^{2} + x - 6 > 0$ are the values of x where the curve is above the x-axis. This happens when x <-3 or x> 2. In set notation: {x: x <-3} ∪ {x: x> 2}

Inequalities Maths Solving quadratic inequalities graph curve above x-axis StudySmarter Solving quadratic inequalities graph - curve above the x-axis, Marilú García De Taylor - StudySmarter Originals

If you want to find the solution for $x^{2} + x - 6 < 0$ , it will be the values of x where the curve is below the x-axis. This happens when -3 <x <2. In set notation: {x: -3 <x <2}

Inequalities Maths Solving quadratic inequalities graph curve below x-axis StudySmarter Solving quadratic inequalities graph - curve below the x-axis, Marilú García De Taylor - StudySmarter Originals

How do you represent inequalities graphically?

You might need to represent the solution to inequalities graphically by considering the Graphs that they relate to.

The rules that apply in this case are:

The values of x for which the curve y = f (x) is below the curve y = g (x) satisfy the inequality f (x) <g (x)
The values of x for which the curve y = f (x) is above the curve y = g (x) satisfy the inequality f (x)> g (x)

Examples of representing inequalities graphically

Given the Equations y = 3x + 10, and $y = x^{2}$ , find the solution for the inequality $3 x + 10 > x^{2}$

Make the equations equal to each other to find the points of intersection and the critical values:

$3 x + 10 = x^{2}$

$x^{2} - 3 x - 10 = 0$ factorise to find the critical values

$(x + 2) (x - 5)$

The critical values are x = -2 and x = 5

Substitute the critical values into $y = x^{2}$ to find the points of intersection :

When x = -2, $y = {(- 2)}^{2} = 4$ A = (- 2, 4)

When x = 5, $y = {(5)}^{2} = 25$ B = (5, 25)

Inequalities Maths Representing inequalities graphically intersection points StudySmarter Representing inequalities graphically - points of intersection, Marilú García De Taylor - StudySmarter Originals

The solution for $3 x + 10 > x^{2}$ are the values of x for which the graph of 3x + 10 is above the graph of $x^{2}$ . This happens when -2 <x <5. In set notation: {x: -2 <x <5}

Representing regions in inequalities

Sometimes when you are working with inequalities, you will be asked to find and shade the region that satisfies linear and quadratic inequalities at the same time.

The best way to approach this type of problem is to represent all the inequalities graphically to find the region where all inequalities are satisfied, paying special consideration to the following guidance:

If the inequalities include the symbols < or >, then the curve is not included in the region, and it needs to be represented with a dotted line.
If the inequalities include the symbols $\leq$ or $\geq$ , then the curve is included in the region, and it needs to be represented with a solid line.

Example of representing regions in inequalities

Shade the region that satisfies the inequalities:

$y + x < 5$ and $y \geq x^{2} - x - 6$

The inequality y + x <5 uses the < symbol, therefore its graph is represented with a dotted line. The inequality $y \geq x^{2} - x - 6$ uses the $\geq$ symbol, therefore it is represented with a solid line.

The region where both inequalities are satisfied at the same time has been shaded in blue.

Inequalities Maths Representing regions in inequalities graphically StudySmarter Representing regions in inequalities graphically, Marilú García De Taylor - StudySmarter Originals

Inequalities Maths - Key takeaways

Inequalities are algebraic expressions that, instead of representing how two terms are equal to each other, represent how one term is less than, less than or equal, greater than, or greater than or equal than the other.
Inequalities can be manipulated in the same way as equations, but must consider a few extra rules.
When multiplying or dividing inequalities by a negative number, the symbol must be reversed so that the inequality continues to be true.
The solution of an inequality is the set of all real numbers that make the inequality true.
You can use a number line to represent two or more inequalities together, to see more clearly the values that satisfy all inequalities at the same time.
Solving quadratic inequalities can be done by factorising, Completing the Square or using the quadratic formula to find the critical values required to be able to draw the corresponding graph and find the solution.

Frequently Asked Questions about Inequalities Maths

An inequality equation is an algebraic expression that instead of an equal symbol (=), contains the symbols less than (<), less than or equal to (≤), greater than (>), or greater than or equal to (≥).

Inequalities can be solved in a similar way to equations, isolating the variable and combining like terms. The solution of the inequality will be the set of all real numbers that make the inequality true. A few extra rules need to be followed, like reversing the symbol of the inequality when multiplying or dividing by a negative number.

Inequality in Maths represents how one term is less than, less than or equal to, greater than, or greater than or equal to another.

Less than (<), less than or equal to (≤), greater than (>), and greater than or equal to (≥).

The properties of inequalities in Maths are:

1. Addition: If a > b, then a + c > b + c

2. Subtraction: If a > b, then a - c > b - c

3. Multiplication:

If a > b and c > 0, then a x c > b x c

If a > b and c < 0, then a x c < b x c

4. Division:

If a > b and c > 0, then a/c > b/c

If a > b and c < 0, then a/c < b/c

5. Transitive: If a > b and b > c, then a > c

6. Comparison: If a = b + c and c > 0, then a > b

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What are Inequalities in Maths?

Inequalities are algebraic expressions that instead of representing how both sides of an equation are equal to each other, represent how one term is less than, less than or equal, greater than, or greater than or equal than the other.

What is the addition property of Inequalities?

If a > b, then a + c > b + c

What is the multiplication property of Inequalities?

If a > b and c > 0, then a x c > b x c

If a > b and c < 0, then a x c < b x c

What is the transitive property of Inequalities?

If a > b and b > c, then a > c

How to solve quadratic inequalities?

It can be done by factorising, completing the square or using the quadratic formula to find the critical values required to be able to draw the corresponding graph and find the solution.

When representing regions in inequalities, what type of line should be used to represent that the curve is not included in the region?

Dotted line

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