Solve the compound inequality [tex]4 x-7>5[/tex] or [tex]5 x+4 \leq-6[/tex]
A) [tex]x<-2[/tex] or [tex]x \geq 3[/tex]
B) [tex]x \leq-2[/tex] or [tex]x>3[/tex]
C) [tex]-2 \leq x<3[/tex]
D) [tex]x \leq-2[/tex] and [tex]x>3[/tex]
Answer (2)
Solve the first inequality 5"> 4 x − 7 > 5 which gives 3"> x > 3 .
Solve the second inequality 5 x + 4 ≤ − 6 which gives x ≤ − 2 .
Combine the solutions using 'or': x ≤ − 2 or 3"> x > 3 .
The final answer is x ≤ − 2 or 3"> x > 3 , which corresponds to option B: 3}"> x ≤ − 2 or x > 3 .
Explanation
Understanding the problem We are given the compound inequality 5"> 4 x − 7 > 5 or 5 x + 4 ≤ − 6 . We need to solve each inequality separately and then combine the solutions.
Solving the first inequality First, let's solve the inequality 5"> 4 x − 7 > 5 . To isolate x , we add 7 to both sides of the inequality: 5 + 7"> 4 x − 7 + 7 > 5 + 7
12"> 4 x > 12
Now, we divide both sides by 4: \frac{12}{4}"> 4 4 x > 4 12
3"> x > 3
Solving the second inequality Next, let's solve the inequality 5 x + 4 ≤ − 6 . To isolate x , we subtract 4 from both sides of the inequality: 5 x + 4 − 4 ≤ − 6 − 4
5 x ≤ − 10
Now, we divide both sides by 5: 5 5 x ≤ 5 − 10
x ≤ − 2
Combining the solutions The solution to the compound inequality is the union of the solutions to the individual inequalities. Therefore, the solution is 3"> x > 3 or x ≤ − 2 .
Final Answer Comparing our solution to the given options, we see that it matches option B: x ≤ − 2 or 3"> x > 3 .
Examples
Compound inequalities are useful in various real-world scenarios. For example, suppose a company wants to offer a discount to customers who either spend more than 100 or a re u n d er 18 ye a rso l d . T hi sc anb ere p rese n t e d a s a co m p o u n d in e q u a l i t y w h ere x i s t h e am o u n t s p e n t an d y i s t h e a g eo f t h ec u s t o m er . T h e d i sco u n t a ppl i es i f x > 100 or y < 18$. Solving compound inequalities helps businesses target specific customer groups for promotions.
The compound inequality 5"> 4 x − 7 > 5 or 5 x + 4 ≤ − 6 resolves to x ≤ − 2 or 3"> x > 3 . The final answer corresponds to option B. Thus, the solution is x ≤ − 2 or 3"> x > 3 .
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