Question

solve for x.
-(x - 2)(x - 5) < 0
write a compound inequality like 1 < x < 3 or like x < 1 or x > 3.

Explanation:

Step1: Simplify the inequality

First, we can multiply both sides of the inequality \(-(x - 2)(x - 5) \lt 0\) by \(- 1\). Remember that when we multiply or divide an inequality by a negative number, the direction of the inequality sign changes. So we get \((x - 2)(x - 5)\gt0\)

Step2: Find critical points

To solve the inequality \((x - 2)(x - 5)\gt0\), we first find the critical points by setting each factor equal to zero.
For \(x - 2=0\), we get \(x = 2\)
For \(x - 5=0\), we get \(x = 5\)

These critical points \(x = 2\) and \(x = 5\) divide the number line into three intervals: \((-\infty,2)\), \((2,5)\) and \((5,\infty)\)

Step3: Test intervals

• For the interval \((-\infty,2)\), let's choose a test point, say \(x = 0\). Substitute \(x = 0\) into \((x - 2)(x - 5)\): \((0 - 2)(0 - 5)=(- 2)\times(-5) = 10\gt0\). So the inequality holds for \(x\lt2\)
• For the interval \((2,5)\), let's choose a test point, say \(x = 3\). Substitute \(x = 3\) into \((x - 2)(x - 5)\): \((3 - 2)(3 - 5)=(1)\times(-2)=- 2\lt0\). So the inequality does not hold for \(2\lt x\lt5\)
• For the interval \((5,\infty)\), let's choose a test point, say \(x = 6\). Substitute \(x = 6\) into \((x - 2)(x - 5)\): \((6 - 2)(6 - 5)=(4)\times(1) = 4\gt0\). So the inequality holds for \(x\gt5\)

Answer:

\(x\lt2\) or \(x\gt5\)