Solved on Feb 03, 2024

Solve the inequality $3|x+1|-2<4$ and select the correct solution.
A) $-3<x$ or $x<1$ B) $-3<x$ or $x>1$ C) $-3<x<1$ D) $-3<x$ and $x>1$

STEP 1

Assumptions
1. We are solving the inequality $3|x+1|-2<4$ .
2. We will consider two cases for the absolute value: when $x+1 \geq 0$ and when $x+1 < 0$ .

STEP 2

First, we isolate the absolute value term by adding 2 to both sides of the inequality.
$3|x+1|-2+2<4+2$

STEP 3

Simplify the inequality.
$3|x+1|<6$

STEP 4

Next, we divide both sides of the inequality by 3 to isolate the absolute value.
$\frac{3|x+1|}{3}<\frac{6}{3}$

STEP 5

Simplify the inequality.
$|x+1|<2$

STEP 6

Now we consider the two cases for the absolute value:
Case 1: $x+1 \geq 0$
Case 2: $x+1 < 0$

STEP 7

For Case 1, where $x+1 \geq 0$ , the absolute value can be removed without changing the sign.
$x+1<2$

STEP 8

Subtract 1 from both sides of the inequality to solve for $x$ in Case 1.
$x+1-1<2-1$

STEP 9

Simplify the inequality for Case 1.
$x<1$

STEP 10

For Case 2, where $x+1 < 0$ , the absolute value will change the sign of the expression inside it.
$-(x+1)<2$

STEP 11

Distribute the negative sign inside the parentheses.
$-x-1<2$

STEP 12

Add 1 to both sides of the inequality to solve for $x$ in Case 2.
$-x-1+1<2+1$

STEP 13

Simplify the inequality for Case 2.
$-x<3$

STEP 14

Multiply both sides of the inequality by -1 to solve for $x$ in Case 2. Remember that multiplying or dividing by a negative number reverses the inequality sign.
$-1(-x)>-1(3)$

STEP 15

Simplify the inequality for Case 2.
$x>-3$

STEP 16

Combine the results from Case 1 and Case 2. The solution to the inequality is the union of the two cases.
$-3<x<1$

STEP 17

The solution to the inequality $3|x+1|-2<4$ is $-3<x<1$ , which corresponds to option C.
C) $-3<x<1$

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Solve the inequality $3|x+1|-2<4$ and select the correct solution.
A) $-3<x$ or $x<1$ B) $-3<x$ or $x>1$ C) $-3<x<1$ D) $-3<x$ and $x>1$

STEP 1

Assumptions
1. We are solving the inequality $3|x+1|-2<4$ .
2. We will consider two cases for the absolute value: when $x+1 \geq 0$ and when $x+1 < 0$ .

STEP 2

First, we isolate the absolute value term by adding 2 to both sides of the inequality.
$3|x+1|-2+2<4+2$

STEP 3

Simplify the inequality.
$3|x+1|<6$

STEP 4

Next, we divide both sides of the inequality by 3 to isolate the absolute value.
$\frac{3|x+1|}{3}<\frac{6}{3}$

STEP 5

Simplify the inequality.
$|x+1|<2$

STEP 6

Now we consider the two cases for the absolute value:
Case 1: $x+1 \geq 0$
Case 2: $x+1 < 0$

STEP 7

For Case 1, where $x+1 \geq 0$ , the absolute value can be removed without changing the sign.
$x+1<2$

STEP 8

Subtract 1 from both sides of the inequality to solve for $x$ in Case 1.
$x+1-1<2-1$

STEP 9

Simplify the inequality for Case 1.
$x<1$

STEP 10

For Case 2, where $x+1 < 0$ , the absolute value will change the sign of the expression inside it.
$-(x+1)<2$

STEP 11

Distribute the negative sign inside the parentheses.
$-x-1<2$

STEP 12

Add 1 to both sides of the inequality to solve for $x$ in Case 2.
$-x-1+1<2+1$

STEP 13

Simplify the inequality for Case 2.
$-x<3$

STEP 14

Multiply both sides of the inequality by -1 to solve for $x$ in Case 2. Remember that multiplying or dividing by a negative number reverses the inequality sign.
$-1(-x)>-1(3)$

STEP 15

Simplify the inequality for Case 2.
$x>-3$

STEP 16

Combine the results from Case 1 and Case 2. The solution to the inequality is the union of the two cases.
$-3<x<1$

STEP 17

The solution to the inequality $3|x+1|-2<4$ is $-3<x<1$ , which corresponds to option C.
C) $-3<x<1$

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Solve the inequality 3∣x+1∣−2<43|x+1|-2<43∣x+1∣−2<4 and select the correct solution.A) −3<x-3<x−3<x or x<1x<1x<1 B) −3<x-3<x−3<x or x>1x>1x>1 C) −3<x<1-3<x<1−3<x<1 D) −3<x-3<x−3<x and x>1x>1x>1

STEP 1

Assumptions1. We are solving the inequality 3∣x+1∣−2<43|x+1|-2<43∣x+1∣−2<4.2. We will consider two cases for the absolute value: when x+1≥0x+1 \geq 0x+1≥0 and when x+1<0x+1 < 0x+1<0.

STEP 2

First, we isolate the absolute value term by adding 2 to both sides of the inequality.3∣x+1∣−2+2<4+23|x+1|-2+2<4+23∣x+1∣−2+2<4+2

STEP 3

Simplify the inequality.3∣x+1∣<63|x+1|<63∣x+1∣<6

STEP 4

Next, we divide both sides of the inequality by 3 to isolate the absolute value.3∣x+1∣3<63\frac{3|x+1|}{3}<\frac{6}{3}33∣x+1∣​<36​

STEP 5

Simplify the inequality.∣x+1∣<2|x+1|<2∣x+1∣<2

STEP 6

Now we consider the two cases for the absolute value:Case 1: x+1≥0x+1 \geq 0x+1≥0Case 2: x+1<0x+1 < 0x+1<0

STEP 7

For Case 1, where x+1≥0x+1 \geq 0x+1≥0, the absolute value can be removed without changing the sign.x+1<2x+1<2x+1<2

STEP 8

Subtract 1 from both sides of the inequality to solve for xxx in Case 1.x+1−1<2−1x+1-1<2-1x+1−1<2−1

STEP 9

Simplify the inequality for Case 1.x<1x<1x<1

STEP 10

For Case 2, where x+1<0x+1 < 0x+1<0, the absolute value will change the sign of the expression inside it.−(x+1)<2-(x+1)<2−(x+1)<2

STEP 11

Distribute the negative sign inside the parentheses.−x−1<2-x-1<2−x−1<2

STEP 12

Add 1 to both sides of the inequality to solve for xxx in Case 2.−x−1+1<2+1-x-1+1<2+1−x−1+1<2+1

STEP 13

Simplify the inequality for Case 2.−x<3-x<3−x<3

STEP 14

Multiply both sides of the inequality by -1 to solve for xxx in Case 2. Remember that multiplying or dividing by a negative number reverses the inequality sign.−1(−x)>−1(3)-1(-x)>-1(3)−1(−x)>−1(3)

STEP 15

Simplify the inequality for Case 2.x>−3x>-3x>−3

STEP 16

Combine the results from Case 1 and Case 2. The solution to the inequality is the union of the two cases.−3<x<1-3<x<1−3<x<1

STEP 17

The solution to the inequality 3∣x+1∣−2<43|x+1|-2<43∣x+1∣−2<4 is −3<x<1-3<x<1−3<x<1, which corresponds to option C.C) −3<x<1-3<x<1−3<x<1

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Solve the inequality $3|x+1|-2<4$ and select the correct solution.
A) $-3<x$ or $x<1$ B) $-3<x$ or $x>1$ C) $-3<x<1$ D) $-3<x$ and $x>1$

Assumptions
1. We are solving the inequality $3|x+1|-2<4$ .
2. We will consider two cases for the absolute value: when $x+1 \geq 0$ and when $x+1 < 0$ .

First, we isolate the absolute value term by adding 2 to both sides of the inequality.
$3|x+1|-2+2<4+2$

Simplify the inequality.
$3|x+1|<6$

Next, we divide both sides of the inequality by 3 to isolate the absolute value.
$\frac{3|x+1|}{3}<\frac{6}{3}$

Simplify the inequality.
$|x+1|<2$

Now we consider the two cases for the absolute value:
Case 1: $x+1 \geq 0$
Case 2: $x+1 < 0$

For Case 1, where $x+1 \geq 0$ , the absolute value can be removed without changing the sign.
$x+1<2$

Subtract 1 from both sides of the inequality to solve for $x$ in Case 1.
$x+1-1<2-1$

Simplify the inequality for Case 1.
$x<1$

For Case 2, where $x+1 < 0$ , the absolute value will change the sign of the expression inside it.
$-(x+1)<2$

Distribute the negative sign inside the parentheses.
$-x-1<2$

Add 1 to both sides of the inequality to solve for $x$ in Case 2.
$-x-1+1<2+1$

Simplify the inequality for Case 2.
$-x<3$

Multiply both sides of the inequality by -1 to solve for $x$ in Case 2. Remember that multiplying or dividing by a negative number reverses the inequality sign.
$-1(-x)>-1(3)$

Simplify the inequality for Case 2.
$x>-3$

Combine the results from Case 1 and Case 2. The solution to the inequality is the union of the two cases.
$-3<x<1$

The solution to the inequality $3|x+1|-2<4$ is $-3<x<1$ , which corresponds to option C.
C) $-3<x<1$