Problem 85 - Algebra and Trigonometry

Question

Find all values of \(x\) satisfying the given conditions. $$y=|5-4 x| \text { and } y=11$$

Step-by-Step Solution

Answer

The solutions for \(x\) are -1.5 and 4.

Step 1: Set up the Equation

The very first step is to set the given equations equal to each other. This gives us: \( |5-4x| = 11 \)

Step 2: Solve for the Positive Case

The definition of absolute value is that it gives the 'magnitude' of a number, which means it could either be positive or negative. Let's first solve for the positive case, which means: \(5 - 4x = 11\). By subtracting 5 from both sides, you end up with \(-4x = 6\). If you then divide all terms by -4 to isolate \(x\), you'll find that \(x = -1.5\) for this case.

Step 3: Solve for the Negative Case

Now, let's solve for the negative case. This time, the equation is \(-(5 - 4x) = 11\). If you distribute the negative sign, it changes the equation to \(-5 + 4x = 11\). Adding 5 to both sides gives you \(4x = 16\). Then, dividing all terms by 4 to isolate \(x\), you will find that \(x = 4\) for this case.