SAT Inequalities Concepts, Strategies & Properties

Understanding Inequalities

In the realm of Algebra, an inequality is a statement that illustrates the relationship between two expressions using an inequality symbol. The expressions on either side of an inequality sign are not equal. This implies that the expression on the left-hand side should be either greater or less than the expression on the right-hand side, or the other way round. When the relationship between two algebraic expressions is depicted using inequality symbols, it is referred to as literal inequalities.

Definition: “ An inequality is a relation where two real numbers or algebraic expressions are related by the symbols “>”, “<”, “≥”, “≤”. .”

For instance, consider x>3 (x should be greater than 3)

Open Sentence: An inequality is referred to as an open sentence if it consists of only one variable.

For instance, x < 6 (x is less than 6)

Double Inequalities : An inequality is said to be a double inequality if the statement displays the double relation of the expressions or the numbers.

For example: 3≤x<8 ( x is greater than or equal to 3 and less than 8)

Further reading : Quadratic Inequalities

Symbols of Inequality

The most common inequality sign is the “not equal sign (≠)”. However, to compare values on the inequalities, the following symbols are used.

Strict Inequality

The strict inequality symbols are less than symbol (<) and greater than symbol (>). These two symbols are referred to as strict inequalities, as they indicate the numbers are strictly greater or less than each other.

For instance,

• 5 < 9 ( 5 is strictly less than 9)
• 10 > 7 (10 is strictly greater than 7)

Slack Inequality

The slack inequalities are less than or equal to symbol (≤) and greater than or equal to symbol (≥). The slack inequalities represent the relation between two inequalities that are not strict.

For instance,

• x ≥ 15 ( x is greater than or equal to 15)
• x ≤ 9 (x is less than or equal to 9)

Properties of Inequalities

The following are the properties of the inequalities:

Transitive Property

The transitive property defines the relationship between three numbers.

If a, b and c are the three numbers, then:

If a ≥ b, and b ≥ c, then a ≥ c

Similarly,

If a ≤ b, and b ≤ c, then a ≤ c

In the examples mentioned above, if one relation is defined by strict inequality, then the result should also be in strict inequality.

For instance,

If a ≥ b, and b > c, then a > c.

Properties of Addition and Subtraction

The properties of addition and subtraction of inequalities state that adding or subtracting the same constant on both sides of inequalities results in equivalent inequalities.

Let “m” be a constant,

If x ≤ y, then x +m ≤ y+m

If x ≥ y, then x +m ≥ y+m

Similarly, for the subtraction operation,

If x ≤ y, then x -m ≤ y-m

If x ≥ y, then x – m ≥ y-m

Properties of Multiplication and Division

If an inequality is multiplied or divided by a positive constant number, the inequality remains unchanged. But, if an inequality is multiplied or divided by a negative constant number, the inequality expression will get reversed.

Let “m” be a positive constant,

If x ≤ y, then xm ≤ ym (if m>0)

If x ≥ y, then xm ≥ ym (if m>0)

Let “m” be a negative constant number,

If x ≤ y, then xm ≥ ym (if m<0)

If x ≥ y, then xm ≤ ym (if m<0)

The above condition holds true for the division operation as well.

Converse Property

The converse property states that if we flip the number, we have to flip the inequality symbol as well.

i.e., If a ≥ b, then b ≤ a

Similarly, if a ≤ b, then b ≥ a.

Solving Inequalities

The process of solving inequalities is quite similar to solving an equation. While solving inequalities, the following rules should be adhered to, as they do not affect the inequality direction:

• Add or subtract the same number on both sides of an inequality.
• Multiply or divide the inequality by the same positive number.
• Simplify a side of the inequality if possible.

Now, let's explore how to solve linear inequalities in one variable and two variables.

Solving Linear Inequalities in One Variable

If the linear inequality contains only one variable, then it is referred to as a linear inequality in one variable. Here, our goal is to find the value of the unknown variable.

Inequalities Example 1:

Solve the linear inequality in one variable: 7x+3<5x+9

Solution:

The given inequality is 7x+3<5x+9.

Subtract 5x on both sides of the inequality.

Thus,

⇒ 7x+3-5x < 5x+9-5x

⇒2x+3 <9

⇒2x < 9-3

⇒ 2x < 6

⇒ x < 3

Hence, the simplified form of the linear inequality 7x+3 < 5x+9 is x < 3.

Solving Linear Inequalities in Two Variables

If the linear inequality contains two variables, then it is known as linear inequality in two variables . In this case, we have to find the solution set for the pair of values of x and y, i.e., (x, y).

Inequalities Example 2:

Solve the linear inequality in two variables: 40x+20y ≤ 120

Solution:

The given inequality is 40x+20y ≤ 120 ….(1)

First, take the L.H.S of an equation, 40x+20y

Let x=0, we get

40x+20y = 40(0)+20y

= 20y

Hence, 20y ≤ 120

y ≤ 6 … (2)

Thus, if x=0, then y can take the values 0, 1, 2, 3, 4, 5, 6.

i.e., (0, 0), (0, 1), (0, 2), (0, 3), (0, 4), (0, 5), (0,6)

Similarly, if we take x = 1, 2, and 3, the possible solutions are:

(1, 0), (1, 1), (1, 2), (1, 3), (1, 4), (2, 0), (2, 1), (2, 2), (3, 0).

Graphing Inequalities

When graphing the linear inequality, we should first solve for the variable. Then, find the solution set, which will assist in graphing the given inequality. Let's learn how to graph linear inequalities in one variable and two variables.

Graphing Linear Inequality in One Variable

If the linear inequality has only one variable, then the graph can be drawn using the number line. Consider example 1 provided above,

i.e., 7x+3<5x+9

The solution set for the inequality is x <3, which is equal to (-∞, 3)

Thus, it can be drawn using the number line.

Graphing Linear Inequality in Two Variables

The graph of the linear inequality in two variables should be on the cartesian plane or xy coordinate plane.

Consider an inequality as discussed above:

40x+20y ≤ 120

To graphically represent this equation, first, consider the equation 40x+20y = 120 and draw its graph.

Now, plot the points (0, 0), (0, 1), (0, 2), (0, 3), (0, 4), (0, 5), (0,6), (1, 0), (1, 1), (1, 2), (1, 3), (1, 4), (2, 0), (2, 1), (2, 2), (3, 0) in the coordinate plane.

To draw the graph of inequality, take one point, say (0, 0) and check whether the point will satisfy the inequality, and we observe that the point (0, 0) satisfies the equation. Hence, plot the remaining points in the plane, and we get the inequality graph as follows:

Practice Problems

Solve the following inequality problems:

1. Solve the inequality 4x+5 < 6x+9
2. Draw the graph for inequality x≤5.
3. Solve the inequality y ≤ x+2 and represent it graphically.

Conclusion

In summary, inequalities are a basic concept in algebra that transcends simple comparisons of numbers to the solution of real-life problems that have constraints, optimization, and decision-making. Their usage is important not only in high school mathematics but also in most competitive tests such as the SAT, ACT, GRE, and GMAT, where problem-solving and logical reasoning are of particular importance. Through the control of inequalities, students are able to better interpret mathematical relationships, solve complicated questions in a cost-effective manner, and enhance their performance in standard tests and otherwise.