Math

Question Create a compound inequality using digits 1-9 (at most once each) that is equivalent to 2x<42 \leq x < 4. Explain your solution.

Studdy Solution

STEP 1

Assumptions
1. We need to use each of the digits from 1 to 9 at most one time each.
2. We are creating a compound inequality equivalent to 2x<42 \leq x < 4.
3. The compound inequality will consist of two parts: one part with "\leq" and another with "<<".

STEP 2

Let's start by constructing the first part of the compound inequality which involves "\leq". We need a number that is equal to or greater than 2. Since we can use each digit only once, we can use the digit '2' itself for this part.
axa \leq x
where a=2a = 2.

STEP 3

Now, let's construct the second part of the compound inequality which involves "<<". We need a number that is less than 4. Since we can use each digit only once and we have already used '2', we can use '3' for this part.
x<bx < b
where b=3b = 3.

STEP 4

Combine the two parts from STEP_2 and STEP_3 to form the compound inequality.
ax<ba \leq x < b

STEP 5

Substitute the values of aa and bb into the compound inequality.
2x<32 \leq x < 3

STEP 6

We need to defend our answer. The compound inequality we have created, 2x<32 \leq x < 3, is not equivalent to the original inequality 2x<42 \leq x < 4 because the upper bound is different. We need to find a way to use the digits 1 to 9 to create an upper bound that is less than 4 but not equal to 3.

STEP 7

To create an upper bound that is less than 4 but not equal to 3, we can use the digits '3' and '9' to form the decimal '3.9'. This gives us a number that is less than 4 but greater than any integer between 2 and 4.
x<3.9x < 3.9

STEP 8

Combine the inequality from STEP_2 with the new upper bound from STEP_7 to form the final compound inequality.
2x<3.92 \leq x < 3.9

STEP 9

We have now created a compound inequality using the digits 1 to 9 at most one time each. The inequality 2x<3.92 \leq x < 3.9 is equivalent to the original inequality 2x<42 \leq x < 4 because it includes all real numbers from 2 up to but not including 4.

STEP 10

Defend the answer by explaining that the compound inequality 2x<3.92 \leq x < 3.9 satisfies the original condition of 2x<42 \leq x < 4 because every number that satisfies 2x<3.92 \leq x < 3.9 will also satisfy 2x<42 \leq x < 4. Additionally, we have used each digit from 1 to 9 at most one time each, as required.
The equivalent compound inequality is 2x<3.92 \leq x < 3.9.

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