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One of the key ways to understand absolute value inequalities is through their graphical representation. When we consider the inequality \[ |4-x| \geq 5 \],we are looking at the graph of the function \[ y = |4-x| \].The absolute value function translates to a V-shaped graph. The vertex of this V is at the point (4,0) because that is where \( 4-x = 0 \).
Now, we need to determine where this graph is greater than or equal to 5 on the y-axis. This means we are looking for the parts of \( y = |4-x| \)that are at or above the horizontal line \( y = 5 \). The intersection points of \( y = |4-x| \) and \( y = 5 \) are the critical values that help solve the inequality. The line is above \( y = 5 \)when \( x \) is less than -1 or greater than 9. This illustrates where the absolute difference from 4 is at least 5.
Visualizing this on a graph offers a powerful way to see the solution: any values of \( x \)that are left of -1 or right of 9 satisfy the inequality.

Solving inequalities involving absolute values might seem tricky at first, but they follow clear logical steps. The core principle is that the absolute value \( |a-b| \) represents the distance between \( a \) and \( b \) on a number line. For \[ |4-x| \geq 5 \], we set up two scenarios because absolute value captures both positive and negative distances:

• \( 4 - x \geq 5 \)
• \( 4 - x \leq -5 \)

Each inequality is solved separately. For \( 4 - x \geq 5 \), solve by subtracting 4 from both sides to get \( -x \geq 1 \). Multiply by \( -1 \) (reversing the inequality sign) to find \( x \leq -1 \). For \( 4 - x \leq -5 \), subtracting 4 gives \( -x \leq -9 \). Again, multiply by \( -1 \) to reverse the inequality and thus \( x \geq 9 \). By breaking down the problem into familiar algebraic steps, you create a clear path to the solution.

The solution set is crucial because it tells us all possible values of \( x \) that satisfy the inequality. After solving individually, we combine the results to form the complete solution set for \[ |4-x| \geq 5 \].
From solving, we get:

• \( x \leq -1 \)
• \( x \geq 9 \)

Together they form the solution set \( x \leq -1 \cup x \geq 9 \).This indicates that any \( x \)value in these ranges will satisfy the original inequality.
In essence, this means the values of \( x \) that are either less than or equal to -1, or greater than or equal to 9, both satisfy the condition where the absolute difference from 4 is at least 5. This visual and algebraic combination ensures full understanding of how solution sets reflect the initial conditions of the problem.
Recognizing the solution set is an important skill, connecting the algebraic solution to practical, visual real-world problems.