Problem 66 - Algebra for College Students

Question

Solve each equation or inequality. Graph the solution set. $$ |-4+x| \leq 9 $$

Step-by-Step Solution

Answer

The solution set is \( -5 \leq x \leq 13 \). On a number line, shade the region between -5 and 13, inclusive.

Step 1: Understand the absolute value inequality

The given inequality is \( |-4+x| \leq 9 \). An absolute value inequality of the form \( |A| \leq B \) means that \(-B \leq A \leq B\).

Step 2: Rewrite the inequality without the absolute value

Rewrite the given inequality \( |-4+x| \leq 9 \) as two separate inequalities: \( -9 \leq -4 + x \leq 9 \).

Step 3: Solve the compound inequality

Solve the compound inequality \( -9 \leq -4 + x \leq 9 \) by isolating \( x \). Add 4 to all parts to get: \( -9 + 4 \leq x \leq 9 + 4 \). Simplifying this, we get \( -5 \leq x \leq 13 \).

Step 4: Graph the solution set

On a number line, mark and shade the region between -5 and 13, inclusive (both -5 and 13 are included in the solution set).

Key Concepts

solving inequalities

When dealing with inequalities, we aim to find all possible values of a variable that make the inequality true.
Think of inequalities as similar to equations, but instead of equality, they show a range of solutions.
To solve an absolute value inequality, start by breaking it down into simpler parts.
For example, take the inequality \(|-4+x| \leq 9\).
This can be rewritten without the absolute value as \( -9 \leq -4 + x \leq 9 \).
The next step is to isolate the variable \( x \).
Add 4 to all parts of the inequality to simplify: \( -9 + 4 \leq x \leq 9 + 4 \).
This resolves to \( -5 \leq x \leq 13 \).
You've now found the values that satisfy the inequality.
This range, from -5 to 13, is the solution set.

graphing solutions

Graphing solutions is a visual way to represent the range of values that satisfy an inequality.
For the inequality \( -5 \leq x \leq 13 \), you'll use a number line.
Start by marking -5 and 13 on the line.
These points include all the values in between them.
Because the inequality includes -5 and 13 (indicated by \( \leq \)), draw solid circles at these points.
Then, shade the region on the number line between -5 and 13.
This shaded area represents all the values that make the inequality true. Graphing helps you understand the solution set intuitively and makes it easier to see the range of solutions at a glance.
Remember, for inequalities that use \( < \) or \( > \), you'd use open circles to indicate those points are not included in the solution set.

compound inequalities

Compound inequalities are sets of inequalities combined together, showing a range of solutions.
They can look complex, but breaking them into manageable steps helps.
Consider the given example, \( -9 \leq -4 + x \leq 9 \).
This is a compound inequality because it combines two inequalities: \( -9 \leq -4 + x \) and \( -4 + x \leq 9 \).
Solving them involves treating them as one.
Start by isolating \( x \) in the middle.
Add 4 to each part: \( -9 + 4 \leq -4 + x + 4 \leq 9 + 4 \), leading to \( -5 \leq x \leq 13 \).
This shows that \( x \) must be greater than or equal to -5 and less than or equal to 13. Compound inequalities help define a precise range where the variable lies.
Practice with these makes the concept clearer, aiding in understanding broader inequality problems.