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

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

Inequalities - Examples, Exercises and Solutions

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

Practice Inequalities

Question 1

Solve the following inequality:

$$ 5x+8<9 $$

Question 2

Solve the inequality:

$$ 5-3x>-10 $$

Question 3

Which diagram represents the solution to the inequality below?

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

Question 4

What is the solution to the following inequality?

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

Question 5

Solve the inequality:

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

Examples with solutions for 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

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 #4

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 #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

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

Question 1

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

Question 2

Which inequality is represented by the numerical axis below?

Question 3

When are the following inequalities satisfied?

$$ 3x+4<9 $$

$$ 3 < x+5 $$

Question 4

Find when the inequality is satisfied:

$$ -3x+15<3x<4x+8 $$

Question 5

Which diagram corresponds to the inequality below?

^{$$ 40x+57\leq5x-13\leq25x+7 $$}

What is its solution?

Exercise #6

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

Video Solution

Answer

$3 ≤ x$

Exercise #7

Which inequality is represented by the numerical axis below?

Video Solution

Answer

$-7 < x \leq2$

Exercise #8

When are the following inequalities satisfied?

$3x+4<9$

$3 < x+5$

Video Solution

Answer

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

Exercise #9

Find when the inequality is satisfied:

$-3x+15<3x<4x+8$

Video Solution

Answer

$2.5 < x$

Exercise #10

Which diagram corresponds to the inequality below?

^{$40x+57≤5x-13≤25x+7$}

What is its solution?

Video Solution

Answer

No solution.

Question 1

Find a $$ a $$ so that:

$$ 0 < 8a+4 \leq -a+9 $$

Question 2

which value of X satisfies:

$$ 8x< 3x+9 $$

but does not exist in:

$$ 5x+4<0 $$

Question 3

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

given that $$ 0 < a $$ .

Question 4

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?

Question 5

Writer A writes $\frac{3}{5}$ of the number of pages writer B writes per day. Both of them together write more than 200 pages per day.

What can be said about the number of pages that writer A writes per day?

Exercise #11

Find a $a$ so that:

$0 < 8a+4 \leq -a+9$

Video Solution

Answer

$-\frac{1}{2} < a ≤ \frac{5}{9}$

Exercise #12

which value of X satisfies:

$8x< 3x+9$

but does not exist in:

$5x+4<0$

Video Solution

Answer

$-0.8 \leq x < 1.8$

Exercise #13

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

given that $0 < a$ .

Video Solution

Answer

No solution

Exercise #14

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?

Video Solution

Answer

_{^{The first is greater by four times of the third and more 14}}

Exercise #15

Writer A writes $\frac{3}{5}$ of the number of pages writer B writes per day. Both of them together write more than 200 pages per day.

What can be said about the number of pages that writer A writes per day?

Video Solution

Answer

More than -75