Discover a wealth of knowledge and get your questions answered at IDNLearn.com. Our community is here to provide detailed and trustworthy answers to any questions you may have.
Sagot :
To solve the compound inequality [tex]\(-5 < 4x + 3 \leq 7\)[/tex], we will break it down into two parts and solve each part separately, then combine the results.
### Step 1: Solve [tex]\(-5 < 4x + 3\)[/tex]
1. Start with the inequality: [tex]\(-5 < 4x + 3\)[/tex]
2. Subtract 3 from both sides to isolate the term involving [tex]\(x\)[/tex]:
[tex]\[ -5 - 3 < 4x \][/tex]
3. Simplify the left side:
[tex]\[ -8 < 4x \][/tex]
4. Divide both sides by 4 to solve for [tex]\(x\)[/tex]:
[tex]\[ -8 / 4 < x \Rightarrow -2 < x \][/tex]
### Step 2: Solve [tex]\(4x + 3 \leq 7\)[/tex]
1. Start with the inequality: [tex]\(4x + 3 \leq 7\)[/tex]
2. Subtract 3 from both sides to isolate the term involving [tex]\(x\)[/tex]:
[tex]\[ 4x + 3 - 3 \leq 7 - 3 \][/tex]
3. Simplify the right side:
[tex]\[ 4x \leq 4 \][/tex]
4. Divide both sides by 4 to solve for [tex]\(x\)[/tex]:
[tex]\[ 4 / 4 \leq x \Rightarrow x \leq 1 \][/tex]
### Step 3: Combine the results
Putting the results from the two parts together:
[tex]\[ -2 < x \quad \text{and} \quad x \leq 1 \][/tex]
Therefore, the solution to the inequality [tex]\(-5 < 4x + 3 \leq 7\)[/tex] is:
[tex]\[ -2 < x \leq 1 \][/tex]
Among the given options, the one that correctly represents this solution is:
[tex]\[ \text{C. } x > -2 \text{ and } x \leq 1 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{C} \][/tex]
### Step 1: Solve [tex]\(-5 < 4x + 3\)[/tex]
1. Start with the inequality: [tex]\(-5 < 4x + 3\)[/tex]
2. Subtract 3 from both sides to isolate the term involving [tex]\(x\)[/tex]:
[tex]\[ -5 - 3 < 4x \][/tex]
3. Simplify the left side:
[tex]\[ -8 < 4x \][/tex]
4. Divide both sides by 4 to solve for [tex]\(x\)[/tex]:
[tex]\[ -8 / 4 < x \Rightarrow -2 < x \][/tex]
### Step 2: Solve [tex]\(4x + 3 \leq 7\)[/tex]
1. Start with the inequality: [tex]\(4x + 3 \leq 7\)[/tex]
2. Subtract 3 from both sides to isolate the term involving [tex]\(x\)[/tex]:
[tex]\[ 4x + 3 - 3 \leq 7 - 3 \][/tex]
3. Simplify the right side:
[tex]\[ 4x \leq 4 \][/tex]
4. Divide both sides by 4 to solve for [tex]\(x\)[/tex]:
[tex]\[ 4 / 4 \leq x \Rightarrow x \leq 1 \][/tex]
### Step 3: Combine the results
Putting the results from the two parts together:
[tex]\[ -2 < x \quad \text{and} \quad x \leq 1 \][/tex]
Therefore, the solution to the inequality [tex]\(-5 < 4x + 3 \leq 7\)[/tex] is:
[tex]\[ -2 < x \leq 1 \][/tex]
Among the given options, the one that correctly represents this solution is:
[tex]\[ \text{C. } x > -2 \text{ and } x \leq 1 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{C} \][/tex]
We value your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Thank you for visiting IDNLearn.com. We’re here to provide accurate and reliable answers, so visit us again soon.