Question

Solve the inequality. Then graph the solution. $$|4-x|<5$$

Short answer

The solution to the inequality is -1 < x < 9.

Step-by-step solution

Handling Absolute Value Inequality

The absolute value inequality \(|4-x| < 5\) can be rewritten as two distinct inequalities. The first one is \(4-x < 5\) and the second one is \(- (4-x) < 5\). This step is pretty much the definition of the absolute value: it can be positive or negative, hence we have two inequalities.

Solving the First Inequality

Let's solve the first inequality \(4-x < 5\). From this we get \(x > 4 - 5\), hence \(x > -1\). So we get the solution of the first inequality.

Solving the Second Inequality

Next, solve the second inequality \(- (4-x) < 5\). Simplify to get \(-4 + x < 5\), which simplifies further to \(x < 5 + 4\) hence \(x < 9\). This is the solution for the second inequality.

Intersection of Both Solutions

The final solution to the problem is the intersection of both solutions \(x > -1\) and \(x < 9\). The intersection of these solutions gives the range of values of x that would satisfy the original inequality. Here, it implies that -1 < x < 9.

Graphing

In order to represent this on a graph, draw a number line where the values of x range from -2 to 10. Indicate a circle at -1 and 9. The solution is represented by a line drawn between -1 and 9, exclusive of -1 and 9, since the inequality sign doesn't have equal.