Problem 3 - Intermediate Algebra

Question

Fill in the blanks. \(-6

Step-by-Step Solution

Answer

Compound

Step 1: Identify the Type of Inequality

To determine the type of inequality from the expression \(-6 < x + 1 \leq 1\), observe the inequality's structure. This expression contains two inequality signs: '<' and '≤', which together form a compound inequality. This type of inequality relates a variable to multiple numeric bounds simultaneously.

Step 2: Determine if the Inequality is Compound

A compound inequality is identified by the presence of two inequality signs involving the same variable. In this case, the expression includes both '<' and '≤', indicating that the inequality is indeed compound. Compound inequalities can be of type 'and' which denotes an intersection of solutions.

Step 3: Identify the Specific Type of Compound Inequality

In the given inequality \(-6 < x + 1 \leq 1\), both inequalities are pointing towards an "and" type. This is because the solution set consists of values for \(x\) satisfying both \(-6 < x + 1\) and \(x + 1 \leq 1\) simultaneously. This means the solution set is the intersection of the solutions for each individual inequality.

Key Concepts

Linear Inequality

Linear inequalities are similar to linear equations but instead of an equals sign, they use inequality symbols such as ">", "<", "≤", or "≥". These symbols indicate a range of values that provide solutions to the inequality. For instance, when you have an expression like \[ ax + b < c \]you're dealing with a linear inequality.
This means there are potentially infinite solutions for the variable that fulfill the condition set by the inequality symbols.
A linear inequality can include one variable like above or even two or more variables like \[ x + y \geq z \],br>but what's important here is that any variable remains raised only to the first power. Conceptually, these inequalities can be visualized on a number line (for one variable) or on a graph (for two variables), highlighting all the possible solutions that satisfy the inequality.

And Inequality

An "and" inequality is a special type of compound inequality which means both parts must be true at once.
Consider the inequality \[ -6 < x + 1 \leq 1 \],which includes both "<" and "≤" symbols.
This signifies that the value of x must satisfy both conditions simultaneously, creating an overlapping or intersecting solution set.
In simpler terms, x must make both inequalities true to be a part of the solution set.
When plotted on a number line, the solution set for an "and" inequality typically appears as a segment of the number line where the conditions overlap.
Thus only the values that lie between these overlapping solutions are valid.

Intersection of Solutions

The intersection of solutions refers to the common set of values that satisfy multiple conditions of a compound inequality.
In terms of our example, both \( -6 < x + 1 \)and\( x + 1 \leq 1 \),are conditions applied to the variable that need to be true at the same point.
This means the final solution is the overlap or intersection of the solution sets of the individual inequalities.
This intersection shows us exactly where the values of x satisfy both conditions simultaneously.
On a graph, this is typically where the shaded regions of each inequality overlap.
In this case, the overlapping part presents the set of solutions that engage both inequalities, clarifying the concept of "intersection of solutions" in compound inequalities.