Algebraic Inequalities Study Guide (page 2)
Introduction to Algebraic Inequalities
Mathematics is like checkers in being suitable for the young, not too difficult, amusing, and without peril to the state.
— Plato (C. 424/423— 348/347 B.C.) Classical Greek Philosopher
In this lesson, you'll learn how to solve single-variable and compound inequalities, and how to simplify algebraic inequalities with two variables.
If an equation is what we write to show two quantities that are equal to each other, what can we write to show that two quantities are NOT equal to each other? An inequality. Inequalities can use the less than sign (<), the greater than sign (>), the less than or equal to sign ≤, and the greater than or equal to sign ≥ to compare two quantities. An algebraic inequality is an algebraic expression that contains one of those four signs.
We solve single-variable equations by isolating the variable on one side of the equal sign and its value on the other side. We solve single-variable inequalities in the same way, except that instead of finding an answer that is a single value, our answer is a set of values.
The equation x + 4 = 9 is solved by subtracting 4 from both sides of the equal sign: x + 4 – 4 = 9 – 4, x = 5. The inequality x + 4 < 9 is solved in the same way: Subtract 4 from both sides of the less than sign: x + 4 – 4 < 9 – 4, x < 5. Our answer is x < 5, which means that all values of x that are less than 5 make the inequality true.
We can represent an inequality on a number line. To show x < 5, we put a circle around the number 5, because it is not a part of our answer (our solution is only values of x that are less than 5), and we highlight all of the values to the left of 5, to show that every number that is less than 5 is part of the solution:
There is one important difference between how we solve an equation and how we solve an inequality. When you are solving an inequality, if you multiply or divide both sides of the equation by a negative number, you must flip the inequality symbol. For example, to solve –5x < 25, divide both sides of the inequality by –5. When you do that, switch the inequality symbol from the less than sign to the greater than sign:
- –5x < 25
- x > 5
Why do we switch the symbol? Let's look at some real numbers. We know that –1 is less than 2, and we show that by writing –1 < 2. If we multiply both sides of the inequality by 2, we have –2 < 4, which is also true. The left side of the inequality became twice as small, and the right side of the inequality became twice as large. But what if we were to divide both sides of –1 < 2 by –1. The left side would become 1, and the right side would become –2. However, 1 is greater than –2. So, we must switch the less than sign to a greater than sign, to show that 1 > –2.
Let's look at another example: 3x – 7 ≥ 2. Add 7 to both sides and divide by 3:
- 3x – 7 ≥ 2
- 3x ≥ 9
- x ≥ 3
The number line of this inequality shows a solid circle around 3, because 3 is part of the solution set (since 3 is greater than or equal to 3):
We cannot divide both sides of an inequality by a variable, because we do not know if the variable is positive or negative. The inequalities x2 > 2x and x > 2 are not the same, because x = –3 would be in the solution set of the first inequality, but not the second inequality.
So far, we have seen inequalities where a variable is less than, less than or equal to, greater than, or greater than or equal to a quantity. Sometimes, a variable is between two quantities. For example, if x can be as small as –4 but less than 7, we would write –4 ≤ x < 7. The number –4 is part of the solution because x can be –4, but 7 is not part of the solution because x is less than 7. We can show this inequality on a number line, too. No arrows are highlighted, because the solution set has two boundaries:
An equation and a simple inequality both have left and right sides. Whatever operation we perform on one side, we perform on the other. A compound inequality has three parts, since it has two inequality symbols. Whatever operation we perform on one part, we must perform on all three parts.
To simplify the compound inequality 7 < x + 2 < 9, isolate the variable, x, in the center of the inequality. Do this by subtracting 2 from all three parts of the inequality, which gives us 5 < x < 7.
To simplify the inequality –4 ≤ –2x ≤ 10, we must divide each part of the inequality by –2. Because we are dividing by a negative number, we must change both inequality symbols: –4 ≤ –2x ≤ 10 becomes 2 ≥ x ≥ –5.
Inequalities With Two Variables
When we are given an inequality with both x and y, we simplify it by writing y in terms of x.
Given the inequality 12x + 3y < 6, we subtract 12x from both sides and divide by 3:
- 12x + 3y < 6
- 3y < 6 – 12x
- y < 2 – 4x
Just as with single-variable inequalities, if we multiply or divide both sides of the inequality by a negative number, we must change the inequality symbol. To solve x – 5y ≥ 15 for y in terms of x, we subtract x from both sides, divide by –5, and change the inequality symbol from ≥ to ≤ :
- x – 5y ≥ 15
- –5y ≥ 15 – x
Find practice problems and solutions for these concepts at Algebraic Inequalities Practice Questions.
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