Problem 29 - Prealgebra

Question

Solve each inequality. Then graph the solution on a number line. $$-32 \geq a+(-5)$$

Step-by-Step Solution

Answer

The solution is \\( a \leq -27 \\). Graph: A closed circle at \\( -27 \\) with shading to the left.

Step 1: Understand the Inequality

The given inequality is \(-32 \geq a + (-5)\). This means we need to find the range of values for \( a \) that satisfy this inequality.

Step 2: Isolate the Variable

Add \( 5 \) to both sides of the inequality to isolate the variable \( a \): \[-32 + 5 \geq a\]This simplifies to:\[-27 \geq a\] or equivalently \( a \leq -27 \).

Step 3: Interpret the Solution

The inequality \( a \leq -27 \) means that \( a \) can be any number less than or equal to \( -27 \). This is the solution set of the inequality.

Step 4: Graph the Solution on a Number Line

On a number line, draw a closed circle at \( -27 \) to indicate that \( a \) can be equal to \( -27 \), and shade all the numbers to the left of \( -27 \) to represent \( a \leq -27 \).

Key Concepts

Solving Inequalities

In the exercise provided, you are dealing with the inequality \(-32 \geq a + (-5)\). Solving an inequality involves finding the set of values for the variable that make the inequality true. To solve for \(a\) in this case, you first need to get \(a\) by itself on one side of the inequality. This is just like solving an equation, but with an important consideration: if you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. Luckily, in this exercise, you won't need to do that.

First, you isolate \(a\) by adding 5 to both sides of the inequality: \(-32 + 5 \geq a\). Simplifying the left side gives you \(-27 \geq a\). Rewriting this gives \(a \leq -27\). This means any number equal to or less than \(-27\) satisfies the inequality.

Graphing Inequalities

To graph an inequality, you will visually represent the set of solutions on a number line. The solution \(a \leq -27\) means you need to represent all numbers that are less than or equal to \(-27\). Here's how to do it step-by-step:

• Identify the boundary point as \(-27\). This is your key number.
• Determine if the boundary point is included or not. Because our inequality includes equal to (\(\leq\)), we will represent this with a closed (solid) dot on the number line.

You now have a clear starting point to express the inequality visually on the number line.

Number Line Representation

Visualizing inequalities on a number line is a practical way to showcase the solution set. For \(a \leq -27\), begin by marking the number \(-27\) on the line. Because \(-27\) is part of the solution, due to the "less than or equal to" condition, you plot it with a filled-in circle. Then, to represent all values of \(a\) that are less than \(-27\), shade the line extending to the left of \(-27\).

This shaded part represents all possible solutions. Using a number line makes it easy to comprehend that infinite numbers satisfy the inequality, as long as \(a\) does not exceed \(-27\). This graphical representation shows not just the specific result but the entire spectrum of numbers that fit the condition.