Find when the inequality is satisfied: $$ -3x+15<3x<4x+8 $$

$$ -8 > x $$ o $$ $$$$ 2\frac{1}{2} < x $$

$$ -8 < x $$ o $$ $$$$ 2\frac{1}{2} < x $$

When are the following inequalities satisfied? $$ 3x+4<9 $$ $$ 3 < x+5 $$

$$ x < 1\frac{2}{3} $$,$$ -2 < x $$

$$ x>1\frac{2}{3} $$,$$ -2>x $$

Which diagram corresponds to the inequality below? $$ 40x+57≤5x-13≤25x+7 $$ What is its solution?

$$ $$ <path d="M568 338.25c1.114 0 1.91.356 2.652 1.098.742.742 1.098 1.538 1.098 2.652s-.356 1.91-1.098 2.652c-.742.742-1.538 1.098-2.652 1.098s-1.91-.356-2.652-1.098c-.742-.742-1.098-1.538-1.098-2.652s.356-1.91 1.098-2.652c.742-.742 1.538-1.098 2.652-1.098Zm0-2.5c-1.648 0-3.352.763-4.42 1.83-.816.817-1.454 2.005-1.709 3.255H395.953a1.25 1.25 0 0 0 0 2.5h165.955c.277 1.188.894 2.305 1.673 3.084 1.067 1.068 2.771 1.831 4.419 1.831 1.648 0 3.352-.763 4.42-1.83 1.067-1.068 1.83-2.772 1.83-4.42 0-1.648-.763-3.352-1.83-4.42-1.068-1.067-2.772-1.83-4.42-1.83Z" fill-opacity=".8" paint-order="stroke fill markers"> <path d="M826 338.25c1.114 0 1.91.356 2.652 1.098.742.742 1.098 1.538 1.098 2.652s-.356 1.91-1.098 2.652c-.742.742-1.538 1.098-2.652 1.098s-1.91-.356-2.652-1.098c-.742-.742-1.098-1.538-1.098-2.652s.356-1.91 1.098-2.652c.742-.742 1.538-1.098 2.652-1.098Zm0-2.5c-1.648 0-3.352.763-4.42 1.83-1.067 1.068-1.83 2.772-1.83 4.42 0 1.648.763 3.352 1.83 4.42 1.068 1.067 2.772 1.83 4.42 1.83 1.648 0 3.352-.763 4.42-1.83.778-.78 1.395-1.897 1.672-3.085h166.143a1.25 1.25 0 0 0 0-2.5H832.129c-.255-1.25-.893-2.438-1.71-3.254-1.067-1.068-2.771-1.831-4.419-1.831Z" fill-opacity=".8" paint-order="stroke fill markers"> -2 -1 No solution.

Find a $$ a $$ so that: $$ 0 < 8a+4 ≤ -a+9 $$

$$ $$$$ -\frac{1}{2} < a ≤ \frac{5}{9} $$

$$ a ≥ \frac{5}{9} $$ o $$ a<-\frac{1}{2} $$

$$ 5a+14 < -2x < 3a+8 $$Calculate X in terms of $$ a $$ given that $$ 0 < a $$.

$$ x > -1.5a-4 $$ $$ x < -2.5a-7 $$

Dice 3 numbers. The first half is greater than the second sum by 5. twice the third number and other 7 less than that amount. What can be said about these elements?

The first is greater by four times of the third and more 14

The first is less than the third multiplied by 2 and more 7

The second is greater than the third

Inequalities | Tutorela

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 $F\left(x\right)>G\left(x\right)$ 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 $X$ values.
Using graphs
we will examine at what $X$ values, $Y$ 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).

Detailed explanation

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Test yourself on inequalities!

Solve the following inequality:

$$ 5x+8<9 $$

Practice more now

Simple Inequality Instructions

Example 1

$3X>6$

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

$X>2$

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

Example 2

$-2X>4$

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

$X<-2$

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

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Test your knowledge

Question 1

Solve the inequality:

$$ 5-3x>-10 $$

Question 2

What is the solution to the following inequality?

$$ 10x-4\leq-3x-8 $$

Question 3

Which diagram represents the solution to the inequality below?

$$ 5-8x<7x+3 $$

Solving inequalities using equations:

Given:

$F\left(x\right)=4x−2$

$g\left(x\right)=−3x+5$

Find when

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

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

$4x−2>−3x+5$

Resolve the inequality:

$7X>7$

$x>1$

This means that when

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

Solution of inequality using graphs:

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

$F\left(x\right)=4x−2$

$g\left(x\right)=−3x+5$

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

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

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

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

$F\left(x\right)=4x−2$

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

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

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

$g\left(x\right)=−3x+5$

$F\left(X\right)$

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

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

We note that we are told that the graphs meet at the point where $X=1$
We will examine the graphs and ask when $f\left(X\right)$ Is the blue graph greater than, $g\left(X\right)$ The purple graph?
The answer is when $X>!$
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 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 #2

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 #3

What is the solution to the following inequality?

$10x-4≤-3x-8$

Video Solution

Step-by-Step Solution

In the exercise, we have an inequality equation.

We treat the inequality as an equation with the sign -=,

And we only refer to it if we need to multiply or divide by 0.

$10x-4 ≤ -3x-8$

We start by organizing the sections:

$10x+3x-4 ≤ -8$

$13x-4 ≤ -8$

$13x ≤ -4$

Divide by 13 to isolate the X

$x≤-\frac{4}{13}$

Let's look again at the options we were asked about:

Answer A is with different data and therefore was rejected.

Answer C shows a case where X is greater than $-\frac{4}{13}$ , although we know it is small, so it is rejected.

Answer D shows a case (according to the white circle) where X is not equal to $-\frac{4}{13}$ , and only smaller than it. We know it must be large and equal, so this answer is rejected.

Therefore, answer B is the correct one!

Answer

Exercise #4

Which diagram represents the solution to the inequality below?

$5-8x<7x+3$

Video Solution

Step-by-Step Solution

First, we will move the elements:

$5-8x>7x+3$

$5-3>7x+8x$
$2>15x$

We divide the answer by 13, and we get:

$x > \frac{2}{15}$

Answer

Exercise #5

Solve the inequality:

$8x+a < 3x-4$

Video Solution

Step-by-Step Solution

Solving an inequality equation is just like a normal equation. We start by trying to isolate the variable (X).

It is important to note that in this equation there are two variables (X and a), so we may not reach a final result.

8x+a<3x-4

We move the sections

8x-3x<-4-a

We reduce the terms

5x<-4-a

We divide by 5

x< -a/5 -4/5

And this is the solution!

Answer

$x < -\frac{1}{5}a-\frac{4}{5}$

Do you know what the answer is?

Question 1

Solve the inequality:

$$ 8x+a < 3x-4 $$

Question 2

What is the solution to the inequality shown in the diagram?

Question 3

Which inequality is represented by the numerical axis below?

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