Solve the inequality: $$ 4x + 7 > 19 $$

All values of $$ x $$ are solutions

Which diagram represents the solution to the inequality below? $$ 5-8x<7x+3 $$

Solve the inequality: $$ 5x+b > 2x+3 $$

It cannot be solved. It depends on $$ b $$

Inequality Practice Problems: Linear & Simple Inequalities

Master inequality solving with step-by-step practice problems. Learn greater than, less than rules, sign flipping, and graphical solutions for linear inequalities.

📚Practice Solving Inequalities - Build Your Skills Step by Step

Solve simple inequalities using addition, subtraction, multiplication and division
Master the critical rule of flipping inequality signs when dividing by negative numbers
Practice solving linear inequalities with variables on both sides of the inequality
Learn to identify when F(x) > G(x) using algebraic methods and equations
Interpret inequality solutions using graphs and visual representations
Apply inequality solving skills to real-world mathematical problems and scenarios

Understanding Inequalities

Complete explanation with examples

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

Test your knowledge with 6 quizzes

What is the solution to the following inequality?

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

Examples with solutions for Inequalities

Step-by-step solutions included

Exercise #1

Solve the inequality:

$7 + 2x < 15$

Step-by-Step Solution

To solve the inequality $7 + 2x < 15$ , we start by isolating the variable $x$ .

First, subtract 7 from both sides of the inequality:

$7 + 2x - 7 < 15 - 7$

Simplifying this, we get:

$2x < 8$

Next, divide both sides by 2 to solve for $x$ :

$\frac{2x}{2} < \frac{8}{2}$

Thus, the solution is:

$x < 4$

Answer:

$x < 4$

Exercise #2

Solve the following inequality:

$5x+8<9$

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:

$x<\frac{1}{5}$

Video Solution

Exercise #3

Solve the inequality:

$5-3x>-10$

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:

$5 > x$

Video Solution

Exercise #4

Solve the following inequality:

$3x+4 \leq 10$

Step-by-Step Solution

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

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

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

Answer:

$x \leq 2$

Exercise #5

Solve the following inequality:

$2x-5 > 3$

Step-by-Step Solution

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

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

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

Answer:

$x > 4$

Frequently Asked Questions

Everything you need to know about Inequalities

What is the most important rule when solving inequalities?

The most critical rule is flipping the inequality sign when multiplying or dividing both sides by a negative number. For example, when solving -2x > 4, you divide by -2 and flip the sign to get x < -2.

How do I solve inequalities with variables on both sides?

Follow these steps: 1) Move all variable terms to one side, 2) Move all constant terms to the other side, 3) Combine like terms, 4) Divide by the coefficient of the variable, 5) Remember to flip the sign if dividing by a negative number.

What's the difference between solving equations and inequalities?

While equations have one unique solution, inequalities have a range of solutions. The main difference is using inequality symbols (>, <, ≥, ≤) instead of the equal sign, and remembering to flip the sign when multiplying or dividing by negative numbers.

How do I know when one function is greater than another using graphs?

Look at the graphs and identify where one line is above the other. The function whose graph is higher has greater y-values for those x-values. The intersection point shows where the functions are equal.

Why do we flip the inequality sign with negative numbers?

When you multiply or divide by a negative number, the order of numbers reverses on the number line. For example, 2 > 1, but when multiplied by -1, we get -2 < -1. This maintains the true relationship between the values.

What are the four main inequality symbols and their meanings?

The four symbols are: > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). The 'equal to' versions include the boundary value in the solution set.

How do I solve compound inequalities step by step?

For compound inequalities like a < x < b: 1) Solve each part separately, 2) Apply operations to all three parts simultaneously, 3) Remember to flip signs if multiplying/dividing by negatives, 4) Write the final solution as a range.

Can inequalities have fractions and parentheses like equations?

Yes, inequalities can contain fractions, parentheses, and other complex elements just like equations. Use the same algebraic rules for simplification, but always remember the sign-flipping rule for negative multiplication or division.

Practice by Question Type

Choose your preferred learning style and practice format for fraction problems

Applying the formula Understanding inequality Worded problems

Inequality Practice Problems: Linear & Simple Inequalities

Understanding Inequalities

Practice Inequalities

Examples with solutions for Inequalities

Frequently Asked Questions

What is the most important rule when solving inequalities?

How do I solve inequalities with variables on both sides?

What's the difference between solving equations and inequalities?

How do I know when one function is greater than another using graphs?

Why do we flip the inequality sign with negative numbers?

What are the four main inequality symbols and their meanings?

How do I solve compound inequalities step by step?

Can inequalities have fractions and parentheses like equations?

More Inequalities Questions

Inequalities

Practice by Question Type