Algebra:
Inequalities

1. 
\(2x+5<7\)
 Solution
2. 
\(5-x\le 6\)
 Solution
3. 
\(2(x-1)>3(2x+3)\)
 Solution
4. 
\(3x+2>5\)
 Solution
5. 
\(5-3x\le 3\)
 Solution
6. 
\(\frac{x+3}{2}\ge \frac{2}{3}\)
 Solution
7. 
\({x}^{2}-3x-4>0\)
 Solution
8. 
\(3{x}^{2}+2x>5\)
 Solution
9. 
\({x}^{2}-3x+2>0\)
 Solution
10. 
\(-{2x}^{2}+5x+12<0\)
 Solution

Inequalities - Introduction

Inequalities are all around us. Some of them might be so familiar that you don't even notice them. Consider the following scenarios: the maximum number of people allowed in the elevator, the speed limit on a traffic sign, the minimum test score needed to pass a class, the number of megabytes you can use per month on your cell phone plan, etc. These are all inequalities. You can write them as follows:
1. Number of people allowed in the elevator ≤ 12
2. Maximum miles per hour allowed ≤ 60
3. Score needed to pass the class ≥ 50
4. Number of megabytes of internet usage per month ≤ 2000
Formally, an algebraic inequality is an expression where, instead of the equal sign used in regular equations, one of the following signs is used:
1.
\(<\)
Less than
(For example:
\(2x-1 < 7\)
)
2.
\(\le \)
Less than or equal to
(For example:
\(2x-1 \le 7\)
)
3.
\(>\)
Greater than
(For example:
\(2x-1 > 7\)
)
4.
\(\ge \)
Greater than or equal to
(For example:
\(2x-1 \ge 7\)
)
The solution of an inequality is the set of values of the unknown where the inequality holds true.

Solving Inequalities

The process of solving an inequality is similar to the one of an equation. However, one important note is, if we multiply the whole inequality by a negative number, we have to flip the inequality sign.
Let's solve the following inequality:
\(-5x+24 < 3x-8\)
1. Just like a regular equation, first, subtract
\(-24\)
from both sides of the inequality.
\(-5x+24+(-24)< 3x-8+(-24)\)
\(-5x < 3x-32\)
2. Subtract
\(-3x\)
to both sides.
\(-5x+(-3x) < 3x-32+(-3x)\)
\(-8x < -32\)
3. Divide both sides by
\(8\)
.
\(\frac{-8x}{8} < -\frac{32}{8}\)
\(-x < -4\)
4. Multiply both sides by -1. (Remember: inequality sign must be flipped.)
\(-x\times -1 < -4\times -1\)
\(x > 4\)
Therefore, the solution is
\(x>4\)
. This says that all values greater than
\(4\)
are a valid solution for this inequality.

What's Next

You can also try our other practice problems. Need a full solution to an inequality problem? Try our inequality calculator. Ready to take your learning to the next level with “how” and “why” steps? Sign up for Cymath Plus today.