Inequalities

🏆Practice inequalities

Inequalities are the "outliers" of equations and many of the rules that apply to equations also apply to inequalities.
In terms of writing, the main difference is that instead of the equal sign "=" "=" , we use greater than ">" ">" or less than "<" "<" signs. 

Inequalities can be simple or more complex and also contain fractions, parentheses, and more. 

Another thing that distinguishes inequalities from equations is that equations with one variable have a unique solution. On the contrary, inequalities have a range of solutions. 

Inequalities between linear functions will translate into questions like when F(x)>G(x) F\left(x\right)>G\left(x\right) or vice versa.
We can answer this type of questions in two ways:

  • Using equations
    if the equations of the two functions are given, we will place them in the inequality, solve it, and find the corresponding X X values.
  • Using graphs
    we will examine at what X X values, Y Y values of the function in question are higher or lower than the function in the inequality.

Mathematical inequality symbols explained: X > Y (greater than), X < Y (less than), X ≥ Y (greater than or equal to), and X ≤ Y (less than or equal to).

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Test yourself on inequalities!

einstein

Solve the following inequality:

\( 2x-5 > 3 \)

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Simple Inequality Instructions

Example 1

3X>6 3X>6

In this case, it is a simple inequality. Just like in the equation, we will divide both sides by 3 3 and we will obtain:

X>2 X>2

This is actually the solution. Every X X is greater than 2 2  


Example 2

2X>4 -2X>4

In this case, note that we must divide both sides by a negative number 2 -2 Therefore, the sign must be reversed, that is, we get:

X<2 X<-2

This is actually the solution. Any X X less than 2 -2

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Solving inequalities using equations:

Given:

F(x)=4x2 F\left(x\right)=4x−2

g(x)=3x+5 g\left(x\right)=−3x+5

Find when

F(x)>g(x) F\left(x\right)>g\left(x\right)

Solution:
We replace the equations in the inequality and we get:

4x2>3x+5 4x−2>−3x+5

Resolve the inequality:

7X>7 7X>7

x>1 x>1

This means that when

X<1,F(x)>g(x)F(x)>g(x) X<1,F\left(x\right)>g\left(x\right)F\left(x\right)>g\left(x\right)


Solution of inequality using graphs:

The following figure is given, in which the graphs of the two lines appear:

F(x)=4x2 F\left(x\right)=4x−2

g(x)=3x+5 g\left(x\right)=−3x+5

The two graphs meet at the point X=1 X=1

1 - Inequality

According to the graph, find when f(X)>g(X) f\left(X\right)>g\left(X\right)

Solution:
The first step:
We will identify which graph belongs to which function.

We can see it in the linear equation F(x) F\left(x\right)

F(x)=4x2 F\left(x\right)=4x−2

The slope is positive - the line goes up and its intersection point with the YY axis is 2 −2 .
Therefore, the blue graph will be F(X) F\left(X\right)

Furthermore,
we can see that in the linear equation G(x) G\left(x\right)

The slope is negative: the line goes down and its intersection point with the Y Y axis is 5 5 .
Therefore, the purple graph will be

g(x)=3x+5 g\left(x\right)=−3x+5

F(X) F\left(X\right)

The second step:
We will write next to each graph its name.

We check when f(X)>g(X) f\left(X\right)>g\left(X\right) That is, at what values of X X is the graph of F(x) F\left(x\right)  greater than the graph of \( g\left(X\right) \ ?).
Let's look at the illustration in front of us, this time with the signs:

the illustration with the signs

We note that we are told that the graphs meet at the point where X=1 X=1
We will examine the graphs and ask when f(X) f\left(X\right) Is the blue graph greater than, g(X) g\left(X\right)  The purple graph?
The answer is when X>! X>!
Pay attention, in both directions we arrive at the same answer and not by coincidence.


Examples and exercises with solutions of inequalities

Exercise #1

Solve the inequality:


5-3x>-10

Video Solution

Step-by-Step Solution

Inequality equations will be solved like a regular equation, except for one rule:

If we multiply the entire equation by a negative, we will reverse the inequality sign.

 

We start by moving the sections, so that one side has the variables and the other does not:

-3x>-10-5

-3x>-15

Divide by 3

-x>-5

Divide by negative 1 (to get rid of the negative) and remember to reverse the sign of the equation.

x<5

Answer

5 > x

Exercise #2

Solve the following inequality:

5x+8<9

Video Solution

Step-by-Step Solution

This is an inequality problem. The inequality is actually an exercise we solve in a completely normal way, except in the case that we multiply or divide by negative.

Let's start by moving the sections:

5X+8<9

5X<9-8

5X<1

We divide by 5:

X<1/5

And this is the solution!

 

Answer

x<\frac{1}{5}

Exercise #3

Solve the following inequality:

2x5>3 2x-5 > 3

Step-by-Step Solution

To solve the inequality 2x5>3 2x-5 > 3 , follow these steps:

1. Add 5 to both sides: 2x>8 2x > 8 .

2. Divide both sides by 2: x>4 x > 4 .

Answer

x>4 x > 4

Exercise #4

Solve the inequality:


2x+93 -2x + 9 \leq 3

Step-by-Step Solution

To solve the inequality 2x+93 -2x + 9 \leq 3 , follow these steps:

1. Subtract 9 from both sides:

2x+9939 -2x + 9 - 9 \leq 3 - 9

2. This simplifies to:

2x6 -2x \leq -6

3. Divide each side by -2, remembering to reverse the inequality sign since dividing by a negative number:

x62 x \geq \frac{-6}{-2}

4. Simplifying gives:

x3 x \geq 3

Thus, the solution is x3 x \leq 3 .

Answer

x3 x \geq 3

Exercise #5

Solve the following inequality:

3x+410 3x+4 \leq 10

Step-by-Step Solution

To solve the inequality 3x+410 3x+4 \leq 10 , follow these steps:

1. Subtract 4 from both sides: 3x6 3x \leq 6 .

2. Divide both sides by 3: x2 x \leq 2 .

Answer

x2 x \leq 2

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