Question

solve the following inequality for x. express your answer in algebraic notation. if there is no solution, indicate
o solution.\ if all real numbers solve the inequality, indicate \all real numbers.\

x² - 16 > 0

answer

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Explanation:

Step1: Factor the left - hand side

Use the difference of squares formula \(a^{2}-b^{2}=(a + b)(a - b)\). Here \(a=x\) and \(b = 4\), so \(x^{2}-16=(x + 4)(x - 4)\). The inequality becomes \((x + 4)(x - 4)>0\).

Step2: Find the critical points

Set \((x + 4)(x - 4)=0\). Using the zero - product property \(x+4 = 0\) gives \(x=-4\), and \(x - 4=0\) gives \(x = 4\). These critical points divide the number line into three intervals: \((-\infty,-4)\), \((-4,4)\), and \((4,\infty)\).

Step3: Test the intervals

• For the interval \((-\infty,-4)\), let \(x=-5\). Then \((-5 + 4)(-5 - 4)=(-1)\times(-9)=9>0\).
• For the interval \((-4,4)\), let \(x = 0\). Then \((0 + 4)(0 - 4)=(4)\times(-4)=-16<0\).
• For the interval \((4,\infty)\), let \(x=5\). Then \((5 + 4)(5 - 4)=(9)\times(1)=9>0\).

Answer:

\(x<-4\) or \(x>4\)