Question

Solve the compound inequality 2x − 3 < 7 and 5 − x ≤ 8. A. x ≥ 3 and x < 2 B. x ≥ 3 and x < 5 C. x ≥ −3 and x < 2 D. x ≥ −3 and x < 5

Answer 1

The solution to the compound inequality is
`\(x \geq -3\)` and x < 5, which is represented by option D.

Let's solve the compound inequality step by step:

1. 2x - 3 < 7

Add 3 to both sides:

2x < 10

Divide by 2 (since the coefficient of x is 2 and we want to isolate x):

x < 5

2.
`\(5 - x \leq 8\)`

Subtract 5 from both sides:

`\(-x \leq 3\)`

Multiply by -1 (note that when multiplying or dividing by a negative number, the inequality sign flips):

`\(x \geq -3\)`

So, the solutions to the compound inequality are x < 5 and
`\(x \geq -3\)`.

Now, looking at the answer choices:

C.
`\(x \geq -3\)` and x < 2 – Incorrect

D.
`\(x \geq -3\)` and x < 5 – Correct

Therefore, the correct answer is D.
`\(x \geq -3\) and \(x < 5\)`.

Answer 2

Given:

The compound inequality
`2x-3 < 7` and
`5-x \leq 8`.

To find:

The solution for the given compound inequality

Solution:

We have, compound inequality
`2x-3 < 7` and
`5-x \leq 8`.

For
`2x-3 < 7`,

Add 3 on both sides.

`2x < 7+3`

`2x < 10`

Divide 2 on both sides.

`x< 5` ...(i)

For
`5-x \leq 8`,

Subtract 5 from both sides.

`-x \leq 8-5`

`-x \leq 3`

Divide both sides by -1. So, the sign of inequality is changed.

`x \geq -3` ...(ii)

Using (i) and (ii), we get the solution of given compound inequality as

`x \geq -3` and
`x< 5`

Therefore, the correct option is D.