How do you solve multi-step inequalities with fractions?
To solve multi-step inequalities with fractions, you can follow these general steps:
1. Simplify the inequality by multiplying through to eliminate any denominators. To do this, find the least common denominator (LCD) and multiply each term of the inequality by it.
2. Apply the distributive property if needed to simplify the expression.
3. Combine like terms on both sides of the inequality if applicable.
4. Isolate the variable on one side of the inequality by adding or subtracting terms as necessary.
5. Remember to reverse the inequality sign if you multiply or divide by a negative value.
6. Check your solution by substituting it back into the original inequality and verify if it holds true.
Let's go through an example to illustrate the steps:
Example:
Solve the inequality: 3/4x - 2/3 < 1/2
Step 1: Multiply each term by the LCD, which is 12, to eliminate the denominators:
(12)(3/4x) - (12)(2/3) < (12)(1/2)
9x - 8 < 6
Step 2: Distribute if necessary:
9x - 8 < 6
Step 3: Combine like terms:
9x < 6 + 8
9x < 14
Step 4: Isolate the variable by adding 8 to both sides:
9x + 8 < 14 + 8
9x < 22
Step 5: Divide both sides by 9, remembering to reverse the inequality sign:
x < 22/9
Step 6: Check the solution by substituting it back into the original inequality:
3/4(22/9) - 2/3 < 1/2
66/36 - 2/3 < 1/2
11/6 - 2/3 < 1/2
(22/12) - (8/12) < 1/2
14/12 < 1/2
7/6 < 1/2
Since 7/6 is not less than 1/2, this solution does not hold true. Therefore, the inequality is not satisfied.
In summary, solving multi-step inequalities with fractions involves simplifying, combining like terms, isolating the variable, and checking the solution.