# 代数学: 不等式

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

# 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.