Shoot down the invaders that make the inequality TRUE.
Use LEFT/RIGHT arrow keys to move, SPACE to shoot.
Welcome, aspiring math-letes and intergalactic defenders! You've just played "Inequality Invaders," a game designed by Arcado Games to make learning algebra fun and engaging. While blasting alien ships is exciting, the real victory comes from mastering the mathematical concepts behind the game. This guide will equip you with the knowledge and strategies to not only conquer "Inequality Invaders" but also to confidently tackle inequalities in your academic journey.
At its core, an inequality is a mathematical statement that compares two expressions, indicating that one is not necessarily equal to the other. Unlike equations, which use an equals sign (=) to show that two expressions have the same value, inequalities use special symbols to show relationships like "greater than," "less than," "greater than or equal to," or "less than or equal to."
Example: 5 < 10 (Five is less than ten)
Example: 12 > 7 (Twelve is greater than seven)
Example: x ≤ 3 (x can be 3, 2, 1, 0, -1, etc.)
Example: y ≥ -2 (y can be -2, -1, 0, 1, 2, etc.)
In "Inequality Invaders," your mission is to identify the value of 'x' that makes the presented inequality a true statement. This means you'll be solving for 'x' and then testing the validity of the statement with different numerical values.
The inequalities in the game are primarily one-step linear inequalities. This means they involve a single variable (usually 'x') and require only one operation (addition, subtraction, multiplication, or division) to isolate that variable.
Solving inequalities is very similar to solving equations. You want to isolate the variable (get 'x' by itself) on one side of the inequality symbol. You do this by performing the inverse operation on both sides.
However, there is ONE CRUCIAL DIFFERENCE:
When you multiply or divide both sides of an inequality by a negative number, you MUST reverse the direction of the inequality symbol.
Let's break down each type of one-step inequality with examples relevant to the game:
Goal: To get 'x' alone, subtract the constant from both sides.
Game Example: Find 'x' such that x + 5 < 12
Step 1: Isolate 'x' by subtracting 5 from both sides.
x + 5 - 5 < 12 - 5
Step 2: Simplify.
x < 7
Meaning: Any value of 'x' that is less than 7 will make the original inequality true. This includes 6, 5, 4, etc. If the game presents invaders with values like 6, 5, or 4, those are the correct ones to shoot. If it presents 7, 8, or 9, those are incorrect.
Game Example: Find 'x' such that x + 3 ≥ 10
Step 1: Subtract 3 from both sides.
x + 3 - 3 ≥ 10 - 3
Step 2: Simplify.
x ≥ 7
Meaning: Any value of 'x' that is greater than or equal to 7 will make the original inequality true. This includes 7, 8, 9, etc. In the game, invaders showing 7, 8, 9, etc., would be correct.
Goal: To get 'x' alone, add the constant to both sides.
Game Example: Find 'x' such that x - 4 > 6
Step 1: Isolate 'x' by adding 4 to both sides.
x - 4 + 4 > 6 + 4
Step 2: Simplify.
x > 10
Meaning: Any value of 'x' greater than 10 will make the original inequality true. This includes 11, 12, 13, etc.
Game Example: Find 'x' such that x - 2 ≤ 5
Step 1: Add 2 to both sides.
x - 2 + 2 ≤ 5 + 2
Step 2: Simplify.
x ≤ 7
Meaning: Any value of 'x' less than or equal to 7 will make the original inequality true. This includes 7, 6, 5, etc.
Goal: To get 'x' alone, divide both sides by the coefficient of 'x'. Remember the golden rule!
Game Example: Find 'x' such that 3x < 15
Step 1: Isolate 'x' by dividing both sides by 3 (a positive number, so the symbol stays the same).
3x / 3 < 15 / 3
Step 2: Simplify.
x < 5
Meaning: Any value of 'x' less than 5 (e.g., 4, 3, 2) makes the inequality true.
Game Example (Crucial!): Find 'x' such that -2x ≥ 8
Step 1: Isolate 'x' by dividing both sides by -2 (a negative number). REMEMBER TO FLIP THE INEQUALITY SYMBOL!
-2x / -2 ≤ 8 / -2
Step 2: Simplify.
x ≤ -4
Meaning: Any value of 'x' less than or equal to -4 (e.g., -4, -5, -6) makes the inequality true. This is a common trap in the game and in real-world math!
Goal: To get 'x' alone, multiply both sides by the denominator. Remember the golden rule!
Game Example: Find 'x' such that x / 2 > 4
Step 1: Isolate 'x' by multiplying both sides by 2 (a positive number, so the symbol stays the same).
(x / 2) * 2 > 4 * 2
Step 2: Simplify.
x > 8
Meaning: Any value of 'x' greater than 8 (e.g., 9, 10, 11) makes the inequality true.
Game Example (Crucial!): Find 'x' such that x / -3 < 2
Step 1: Isolate 'x' by multiplying both sides by -3 (a negative number). REMEMBER TO FLIP THE INEQUALITY SYMBOL!
(x / -3) * -3 > 2 * -3
Step 2: Simplify.
x > -6
Meaning: Any value of 'x' greater than -6 (e.g., -5, -4, -3) makes the inequality true.
Now that you understand the math, let's talk about how to apply it effectively in the game to maximize your score and survive longer!
The inequality displayed at the top of the screen is your primary directive. Don't just glance at it. Pay attention to:
The game is fast-paced, so quickly solving the inequality in your head is key. Practice these types of problems outside the game to build your mental math speed. For example, if you see "x + 7 < 15", immediately think "15 - 7 = 8, so x < 8".
This is where many players (and students) make mistakes. Whenever you are multiplying or dividing both sides by a negative number, you MUST flip the inequality symbol. If you see an inequality like "-3x > 9", remember to divide by -3 and flip the sign to "x < -3". Invaders with values like -4, -5, etc., would be correct, while values like -2, -1, 0 would be incorrect.
The game often generates these types of problems, especially at higher levels, to test your understanding of this rule.
After you've solved for 'x', look at the values displayed on the invaders. For each invader, substitute its value into your solved inequality and check if the statement is true. For instance, if you solved for "x < 8":
Your main goal is to shoot the invaders that represent the correct solution. Missing a correct invader or letting it reach the bottom will result in a game over. Incorrect invaders that reach the bottom will penalize your score, but won't end the game (unless your score drops to zero, which isn't implemented in this version, but good practice for other games!).
As the game progresses, you might have multiple correct invaders on screen. Focus on the ones closest to the bottom, or the ones that are easiest to hit without risking a collision with an incorrect one.
While there's no ammo limit, rapid firing can sometimes obscure your view or lead to accidental hits on incorrect invaders if you're not precise. Aim carefully. A well-placed shot on a correct invader is better than a spray-and-pray approach.
The scoring system emphasizes accuracy and speed in identifying the correct solutions. The level progression ensures that your mental math skills are constantly challenged.
Like any skill, mastering inequalities (and "Inequality Invaders") requires practice. The more you play, the faster your brain will process the inequalities and identify the correct solutions. Don't be discouraged by a low score initially; focus on understanding the math behind each problem.
While "Inequality Invaders" focuses on one-step inequalities, the world of algebra extends far beyond. As you progress in math, you'll encounter:
Example: 2x + 5 < 17You would first subtract 5, then divide by 2.
Example: 3x - 4 > x + 8
Example: -3 < x + 1 < 5 (meaning x + 1 is greater than -3 AND less than 5)
Example: |x - 2| > 5
The foundational understanding you gain from solving one-step inequalities is crucial for these more advanced topics. The principles of inverse operations and, critically, the rule about flipping the inequality sign when multiplying or dividing by a negative number, remain constant.
"Inequality Invaders" is more than just a game; it's a dynamic learning tool. By actively engaging with the math problems and applying the strategies outlined in this guide, you'll not only achieve higher scores but also build a solid understanding of algebraic inequalities. Remember to solve, strategize, and shoot accurately! Good luck, and happy invading!