QuestionSolve the inequality $|3p - 4| < p + 8$ and graph the solution set.

Studdy Solution

STEP 1

Assumptions1. The given absolute value inequality is $|3p-4|<p+8$ . We need to solve for $p$
3. We need to graph the solution set

STEP 2

We first need to consider the two cases of the absolute value inequality. The absolute value of a number is less than another number if the original number is between the positive and negative of the other number. So, we have $-p-8<p-4<p+8$

STEP 3

Let's solve the left inequality first. We can start by adding $p$ to both sides to isolate the $3p$ term.
$-8<2p-$

STEP 4

Next, add $4$ to both sides to isolate the $2p$ term.
$-4<2p$

STEP 5

Finally, divide both sides by $2$ to solve for $p$ .
$-2<p$

STEP 6

Now, let's solve the right inequality. We can start by subtracting $p$ from both sides to isolate the $3p$ term.
$3p-4<8$

STEP 7

Next, add $4$ to both sides to isolate the $3p$ term.
$3p<12$

STEP 8

Finally, divide both sides by $3$ to solve for $p$ .
$p<4$

STEP 9

Now, we have two inequalities $-2<p$ and $p<4$ . The solution to the original inequality is the intersection of these two inequalities, which is $-2<p<4$ .

STEP 10

To graph the solution set, we draw a number line and mark the points $-2$ and $4$ . Since $p$ is strictly greater than $-2$ and strictly less than $4$ , we use open circles at $-2$ and $4$ to indicate that these values are not included in the solution set. Then, we shade the region between $-2$ and $4$ to represent all the values of $p$ that satisfy the inequality.
The solution set is $(-2,4)$ .

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QuestionSolve the inequality $|3p - 4| < p + 8$ and graph the solution set.

Studdy Solution

STEP 1

Assumptions1. The given absolute value inequality is $|3p-4|<p+8$ . We need to solve for $p$
3. We need to graph the solution set

STEP 2

We first need to consider the two cases of the absolute value inequality. The absolute value of a number is less than another number if the original number is between the positive and negative of the other number. So, we have $-p-8<p-4<p+8$

STEP 3

Let's solve the left inequality first. We can start by adding $p$ to both sides to isolate the $3p$ term.
$-8<2p-$

STEP 4

Next, add $4$ to both sides to isolate the $2p$ term.
$-4<2p$

STEP 5

Finally, divide both sides by $2$ to solve for $p$ .
$-2<p$

STEP 6

Now, let's solve the right inequality. We can start by subtracting $p$ from both sides to isolate the $3p$ term.
$3p-4<8$

STEP 7

Next, add $4$ to both sides to isolate the $3p$ term.
$3p<12$

STEP 8

Finally, divide both sides by $3$ to solve for $p$ .
$p<4$

STEP 9

Now, we have two inequalities $-2<p$ and $p<4$ . The solution to the original inequality is the intersection of these two inequalities, which is $-2<p<4$ .

STEP 10

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QuestionSolve the inequality ∣3p−4∣<p+8|3p - 4| < p + 8∣3p−4∣<p+8 and graph the solution set.

Studdy Solution

STEP 1

Assumptions1. The given absolute value inequality is ∣3p−4∣<p+8|3p-4|<p+8∣3p−4∣<p+8 . We need to solve for ppp3. We need to graph the solution set

STEP 2

We first need to consider the two cases of the absolute value inequality. The absolute value of a number is less than another number if the original number is between the positive and negative of the other number. So, we have−p−8<p−4<p+8-p-8<p-4<p+8−p−8<p−4<p+8

STEP 3

Let's solve the left inequality first. We can start by adding ppp to both sides to isolate the 3p3p3p term.−8<2p−-8<2p-−8<2p−

STEP 4

Next, add 444 to both sides to isolate the 2p2p2p term.−4<2p-4<2p−4<2p

STEP 5

Finally, divide both sides by 222 to solve for ppp.−2<p-2<p−2<p

STEP 6

Now, let's solve the right inequality. We can start by subtracting ppp from both sides to isolate the 3p3p3p term.3p−4<83p-4<83p−4<8

STEP 7

Next, add 444 to both sides to isolate the 3p3p3p term.3p<123p<123p<12

STEP 8

Finally, divide both sides by 333 to solve for ppp.p<4p<4p<4

STEP 9

Now, we have two inequalities −2<p-2<p−2<p and p<4p<4p<4. The solution to the original inequality is the intersection of these two inequalities, which is −2<p<4-2<p<4−2<p<4.

STEP 10

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QuestionSolve the inequality $|3p - 4| < p + 8$ and graph the solution set.

Assumptions1. The given absolute value inequality is $|3p-4|<p+8$ . We need to solve for $p$
3. We need to graph the solution set

We first need to consider the two cases of the absolute value inequality. The absolute value of a number is less than another number if the original number is between the positive and negative of the other number. So, we have $-p-8<p-4<p+8$

Let's solve the left inequality first. We can start by adding $p$ to both sides to isolate the $3p$ term.
$-8<2p-$

Next, add $4$ to both sides to isolate the $2p$ term.
$-4<2p$

Finally, divide both sides by $2$ to solve for $p$ .
$-2<p$

Now, let's solve the right inequality. We can start by subtracting $p$ from both sides to isolate the $3p$ term.
$3p-4<8$

Next, add $4$ to both sides to isolate the $3p$ term.
$3p<12$

Finally, divide both sides by $3$ to solve for $p$ .
$p<4$

Now, we have two inequalities $-2<p$ and $p<4$ . The solution to the original inequality is the intersection of these two inequalities, which is $-2<p<4$ .