Question Find the interval for the inequality $3 > -x > -4$ .

Studdy Solution

STEP 1

Assumptions
1. We are given the inequality $3 > -x > -4$ .
2. We need to solve for $x$ and express the solution in interval notation.
3. Interval notation for an inequality $a < x < b$ is expressed as $(a, b)$ .
4. We will graph the interval on a number line.

STEP 2

First, we need to solve the inequality for $x$ . We can do this by multiplying the entire inequality by $-1$ to get $x$ by itself. Remember that multiplying by a negative number reverses the inequality signs.
$-1 \cdot 3 < -1 \cdot (-x) < -1 \cdot (-4)$

STEP 3

Now, multiply each part of the inequality by $-1$ .
$-3 < x < 4$

STEP 4

Notice that the inequality signs have been reversed due to multiplication by a negative number. We now have the inequality in terms of $x$ .

STEP 5

The inequality $-3 < x < 4$ can be written in interval notation. The interval notation for this inequality is:
$(-3, 4)$

STEP 6

To graph the interval on a number line, we draw a number line with points at $-3$ and $4$ .

STEP 7

Since the inequality does not include the endpoints (the inequality is strict, not "less than or equal to" or "greater than or equal to"), we use open circles to indicate that $-3$ and $4$ are not included in the interval.

STEP 8

Shade the region between $-3$ and $4$ to represent all the numbers $x$ can take that satisfy the inequality $-3 < x < 4$ .
The interval in interval notation is $(-3, 4)$ .

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Question Find the interval for the inequality $3 > -x > -4$ .

Studdy Solution

STEP 1

Assumptions
1. We are given the inequality $3 > -x > -4$ .
2. We need to solve for $x$ and express the solution in interval notation.
3. Interval notation for an inequality $a < x < b$ is expressed as $(a, b)$ .
4. We will graph the interval on a number line.

STEP 2

First, we need to solve the inequality for $x$ . We can do this by multiplying the entire inequality by $-1$ to get $x$ by itself. Remember that multiplying by a negative number reverses the inequality signs.
$-1 \cdot 3 < -1 \cdot (-x) < -1 \cdot (-4)$

STEP 3

Now, multiply each part of the inequality by $-1$ .
$-3 < x < 4$

STEP 4

Notice that the inequality signs have been reversed due to multiplication by a negative number. We now have the inequality in terms of $x$ .

STEP 5

The inequality $-3 < x < 4$ can be written in interval notation. The interval notation for this inequality is:
$(-3, 4)$

STEP 6

To graph the interval on a number line, we draw a number line with points at $-3$ and $4$ .

STEP 7

Since the inequality does not include the endpoints (the inequality is strict, not "less than or equal to" or "greater than or equal to"), we use open circles to indicate that $-3$ and $4$ are not included in the interval.

STEP 8

Shade the region between $-3$ and $4$ to represent all the numbers $x$ can take that satisfy the inequality $-3 < x < 4$ .
The interval in interval notation is $(-3, 4)$ .

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Question Find the interval for the inequality 3>−x>−43 > -x > -43>−x>−4.

Studdy Solution

STEP 1

Assumptions1. We are given the inequality 3>−x>−43 > -x > -43>−x>−4.2. We need to solve for xxx and express the solution in interval notation.3. Interval notation for an inequality a<x<ba < x < ba<x<b is expressed as (a,b)(a, b)(a,b).4. We will graph the interval on a number line.

STEP 2

STEP 3

Now, multiply each part of the inequality by −1-1−1.−3<x<4-3 < x < 4−3<x<4

STEP 4

Notice that the inequality signs have been reversed due to multiplication by a negative number. We now have the inequality in terms of xxx.

STEP 5

The inequality −3<x<4-3 < x < 4−3<x<4 can be written in interval notation. The interval notation for this inequality is:(−3,4)(-3, 4)(−3,4)

STEP 6

To graph the interval on a number line, we draw a number line with points at −3-3−3 and 444.

STEP 7

Since the inequality does not include the endpoints (the inequality is strict, not "less than or equal to" or "greater than or equal to"), we use open circles to indicate that −3-3−3 and 444 are not included in the interval.

STEP 8

Shade the region between −3-3−3 and 444 to represent all the numbers xxx can take that satisfy the inequality −3<x<4-3 < x < 4−3<x<4.The interval in interval notation is (−3,4)(-3, 4)(−3,4).

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Question Find the interval for the inequality $3 > -x > -4$ .

Assumptions
1. We are given the inequality $3 > -x > -4$ .
2. We need to solve for $x$ and express the solution in interval notation.
3. Interval notation for an inequality $a < x < b$ is expressed as $(a, b)$ .
4. We will graph the interval on a number line.

Now, multiply each part of the inequality by $-1$ .
$-3 < x < 4$

Notice that the inequality signs have been reversed due to multiplication by a negative number. We now have the inequality in terms of $x$ .

The inequality $-3 < x < 4$ can be written in interval notation. The interval notation for this inequality is:
$(-3, 4)$

To graph the interval on a number line, we draw a number line with points at $-3$ and $4$ .

Since the inequality does not include the endpoints (the inequality is strict, not "less than or equal to" or "greater than or equal to"), we use open circles to indicate that $-3$ and $4$ are not included in the interval.

Shade the region between $-3$ and $4$ to represent all the numbers $x$ can take that satisfy the inequality $-3 < x < 4$ .
The interval in interval notation is $(-3, 4)$ .