Solution: Step 1: Isolate the variable y in the compound inequality. - Begin by eliminating terms that do not include y from the middle part of the inequality. Step 1.1: Subtract 2 from each part of the inequality to remove the constant term. - Apply the subtraction to all three parts: -4 - 2 y + 2 - 2 -10 - 2 Step 1.2: Calculate the result of subtracting 2 from -4. - This simplifies the leftmost part of the inequality: -6 y Step 1.3: Calculate the result of subtracting 2 from -10. - This simplifies the rightmost part of the inequality: y -12 - The updated inequality is: -6 y -12 Step 2: Arrange the inequality in the standard form. - Write the inequality with the smallest number on the left and the largest on the right: -12 y -6 Step 3: - The solution is complete, and the interval notation for y is (-12, -6). Knowledge Notes: - Compound inequalities involve expressions with more than one inequality sign. They can often be treated as two separate inequalities that are solved simultaneously. - When solving inequalities, it's important to maintain the order of the terms and to apply any arithmetic operations to all parts of the inequality. - Subtracting or adding the same number to all parts of a compound inequality does not change the direction of the inequality signs. - Multiplying or dividing all parts of an inequality by a negative number reverses the direction of the inequality signs. - Interval notation is a way of writing the set of solutions to an inequality. For example, the interval (-12, -6) represents all numbers between -12 and -6, not including -12 and -6 …

Graph -4> y+2> -10

The question presented is related to inequalities and plotting a region on a two-dimensional coordinate system. Specifically, the inequality -4 > y + 2 > -10 needs to be broken down into two separate linear inequalities and then graphically represented. The problem requires determining the range of y values that satisfy both conditions simultaneously and depicting this range as a shaded region or a line on the graph between y-intercepts derived from the two linear inequalities.

$- 4 > y + 2 > - 10$

Answer

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Solution:

Step 1: Isolate the variable $y$ in the compound inequality.

Begin by eliminating terms that do not include $y$ from the middle part of the inequality.

Step 1.1: Subtract $2$ from each part of the inequality to remove the constant term.

Apply the subtraction to all three parts: $-4 - 2 > y + 2 - 2 > -10 - 2$

Step 1.2: Calculate the result of subtracting $2$ from $-4$.

This simplifies the leftmost part of the inequality: $-6 > y$

Step 1.3: Calculate the result of subtracting $2$ from $-10$.

This simplifies the rightmost part of the inequality: $y > -12$
The updated inequality is: $-6 > y > -12$

Step 2: Arrange the inequality in the standard form.

Write the inequality with the smallest number on the left and the largest on the right: $-12 < y < -6$

Step 3:

The solution is complete, and the interval notation for $y$ is $(-12, -6)$.

Knowledge Notes:

Compound inequalities involve expressions with more than one inequality sign. They can often be treated as two separate inequalities that are solved simultaneously.
When solving inequalities, it's important to maintain the order of the terms and to apply any arithmetic operations to all parts of the inequality.
Subtracting or adding the same number to all parts of a compound inequality does not change the direction of the inequality signs.
Multiplying or dividing all parts of an inequality by a negative number reverses the direction of the inequality signs.
Interval notation is a way of writing the set of solutions to an inequality. For example, the interval $(-12, -6)$ represents all numbers between $-12$ and $-6$, not including $-12$ and $-6$ themselves.
It is crucial to write the interval with the smaller number first, followed by the larger number, to properly represent the solution set.
Inequalities and intervals are fundamental concepts in algebra and are widely used in various fields, including mathematics, economics, engineering, and the sciences.