Question

10 multiple choice 1 point solve the quadratic inequality -x² - 5x + 6 > 0. untitled graph + 10 -10 0 -10 log in or sign up to save your beautiful math! desmos

Explanation:

Step1: Multiply both sides by -1 (reverse inequality)

Multiply the inequality \(-x^{2}-5x + 6>0\) 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 \(x^{2}+5x - 6<0\)

Step2: Factor the quadratic expression

Factor \(x^{2}+5x - 6\). We need two numbers that multiply to \(-6\) and add up to \(5\). The numbers are \(6\) and \(- 1\). So, \(x^{2}+5x - 6=(x + 6)(x - 1)\). The inequality becomes \((x + 6)(x - 1)<0\)

Step3: Find critical points

Set each factor equal to zero: \(x+6 = 0\) gives \(x=-6\) and \(x - 1=0\) gives \(x = 1\). These critical points divide the number line into three intervals: \((-\infty,-6)\), \((-6,1)\), and \((1,\infty)\)

Step4: Test intervals

• For the interval \((-\infty,-6)\), let's pick \(x=-7\). Then \((-7 + 6)(-7 - 1)=(-1)\times(-8) = 8>0\), so the inequality is not satisfied here.
• For the interval \((-6,1)\), let's pick \(x = 0\). Then \((0 + 6)(0 - 1)=(6)\times(-1)=-6<0\), so the inequality is satisfied here.
• For the interval \((1,\infty)\), let's pick \(x=2\). Then \((2 + 6)(2 - 1)=(8)\times(1) = 8>0\), so the inequality is not satisfied here.

Answer:

The solution to the inequality \(-x^{2}-5x + 6>0\) is \(-6 < x < 1\)