First, let's split the compound inequality into separate inequalities.

Compound Inequality : - 9.5 < 3 First Inequality : - 9.5 < 3 Second Inequality : 3 - y \leq 1.3 - y - y \leq 1.3

Notice that compound inequalities written in this way are equivalent to compound inequalities that involve the word and.

- 9.5 < 3 - y and 3 - y \leq 1.3

Let's solve the inequalities separately.

First Inequality

Inequalities can be solved in the same way as equations, by performing inverse operations on both sides until the variable is isolated. The only difference is that when you divide or multiply by a negative number, you must flip the inequality sign.

3 - y \leq 1.3

SubIneq

$LHS - 3 < RHS - 3$

- y \leq - 1.7

DivNegIneq

Divide by $- 1$ and flip inequality sign

y \geq 1.7

This inequality tells us that all values greater than or equal to

1.7

will satisfy the inequality.

Note that the point on $1.7$ is closed because it is included in the solution set.

Second Inequality

Once more, we will solve the inequality by isolating the variable.

- 9.5 < 3 - y

SubIneq

$LHS - 3 < RHS - 3$

- 12.5 < - y

DivNegIneq

Divide by $- 1$ and flip inequality sign

12.5 > y

RearrangeIneq

Rearrange inequality

y < 12.5

This inequality tells us that all values less than

12.5

will satisfy the inequality.

Note that the point on $12.5$ is open because it is not included in the solution set.

Compound Inequality

The solution set to the compound inequality is the intersection of the solution sets. To help visualize the algebraic expression, we will write

y \geq 1.7

as

1.7 \leq y .

First Solution Set : 1.7 \leq y Second Solution Set : y Intersecting Solution Set : 1.7 \leq y < 12.5 < 12.5

Finally, we will graph the solution set to the compound inequality on a number line.

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