What are inequalities?
Inequalities are mathematical expressions that show the relationship between two values when they are not equal. They tell us if one value is less than, greater than, less than or equal to, or greater than or equal to another value.
< means "less than"
> means "greater than"
≤ means "less than or equal to"
≥ means "greater than or equal to"
Inequality properties:
Inequalities have several important properties that are used when solving them. These properties help maintain the inequality's relationship while manipulating the expressions involved.
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Addition and subtraction properties of inequalities: If you add or subtract the same number to both sides of an inequality, the inequality remains true:
- a < b, then a +c < b + c
- a < b, then a - c < b -c
- 3 < 5, then 3 + 4 < 5 + 4
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Multiplication and division of inequalities: If you multiply and divide both sides of an inequality by a positive number, the inequality remains true. If you multiply both sides by a negative number, the inequality sign reverses:
- If a < b and c > 0, then ac < bc
- If a < b and c < 0, then ac > bc
- 3 < 5, then 3 x 4 < 5 x 4
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Transitive Property of Inequalities: If one quantity is less than a second quantity, and the second quantity is less than a third quantity, then the first quantity is less than the third quantity:
- If a < b and b < c, then a < c
- 3 < 5 and 5 < 7, then 3 < 7
Solving inequalities with One Unknown
Example:
3x
-
5
<
7
3x
-
5 + 5
<
7 + 5
3x
<
12
3x ÷ 3
<
12 ÷ 3
x < 4
-
Step 1: Isolate the Variable on One Side.
To isolate the variable x, we need to get rid of the constant term on the side with the variable. Add 5 to both sides
- 3x - 5 + 5 < 7 + 12
- 3x < 12
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Step 2: Solve for the Variable. Now, we need to solve for x by isolating it completely. Divide both sides by 3
- 3x ÷ 3 < 12 ÷ 3
- x < 4
- Step 3: The final result. The result is x < 4
Quizzes:
1. A gardener has at most 100 feet of fencing to enclose a rectangular garden. If the width of the garden is 20 feet, what is the maximum possible length of the garden?
Find the length (L)
2. A baker needs to make at least 50 cookies for an event. If he has already made 18 cookies, how many more does he need to bake?
3. A student needs a minimum average score of 75 to pass a course. If the student's scores on the first three tests are 70, 80, and 85, what score is needed on the fourth test to pass?
4. A car rental company charges a base fee of $25 plus $0.15 per mile driven. If a customer can spend at most $100, how many miles can they drive?
5. A hiker wants to complete a trail that is at least 15 miles long. If they have already hiked 6.5 miles, how much more distance must they cover?