Question

Use the drop-down menus to complete each equation so the statement about its solution is true.

a) 6x–3x<4–2x has no solutions.
b) 6x–3x<4–2x has infinitely many solutions.
c) 6x–3x<4–2x has one solution.
d) 6x–3x<4–2x has three solutions.

Answer 1

To solve the equation 6x - 3x < 4 - 2x for the given conditions, we simplify to 3x < 4 - 2x and then adjust constants to create a contradiction for no solutions, or make both sides equal for infinitely many solutions.

Step-by-step explanation:

The student asked to complete the inequality 6x−3x<4−2x to match the given statements regarding the nature of its solution. The process involves simplifying the inequality and then determining the necessary constants to satisfy the conditions for no solution, infinitely many solutions, one solution, or three solutions.

First, let's simplify the equation:

• 6x − 3x < 4 − 2x
• 3x < 4 − 2x

Now, to address each statement:

1. No solutions: We need to create a contradiction, such as 0x < – some positive number. For example, if we make 4 actually 3x on the left side:
2. 3x < 3x − 2x
3. 3x < x
4. 2x < 0, which is a contradiction and has no solutions.
5. Infinitely many solutions: If we make both sides of the inequality the same, e.g., by having 4 be get no solutions mathematics, an inequality that cannot be true under any circumstances, resulting in no solutions.