Use a sign chart to solve the inequality. Express the answer in inequality and interval notation.

The solution expressed in inequality notation is or .
(Type an inequality or a compound inequality.)
The solution expressed in interval notation is

Question

Use a sign chart to solve the inequality. Express the answer in inequality and interval notation.

The solution expressed in inequality notation is or .
(Type an inequality or a compound inequality.)
The solution expressed in interval notation is

Ask by Herrera Coleman. in the United States

Mar 23,2025

UpStudy AI · Accepted Answer

We begin with the inequality $$ x^2 + 36 > 13x. $$ **Step 1. Rearrange the inequality** Subtract $$13x$$ from both sides to write the inequality in standard form: $$ x^2 - 13x + 36 > 0. $$ **Step 2. Factor the quadratic** We look for two numbers whose product is $$36$$ and whose sum is $$-13$$. These numbers are $$-4$$ and $$-9$$ because $$ (-4)(-9) = 36 \quad \text{and} \quad (-4) + (-9) = -13. $$ Thus, the quadratic factors as: $$ (x - 4)(x - 9) > 0. $$ **Step 3. Identify the critical points** Set each factor equal to zero: $$ x - 4 = 0 \quad \Rightarrow \quad x = 4, $$ $$ x - 9 = 0 \quad \Rightarrow \quad x = 9. $$ These critical points divide the number line into three intervals: - $$x < 4$$, - $$4 < x < 9$$, - $$x > 9$$. **Step 4. Determine the sign on each interval** - For $$x < 4$$: Choose a test point such as $$x = 0$$. Then, $$(0 - 4)(0 - 9) = (-4)(-9) = 36 > 0$$. So the expression is positive. - For $$4 < x < 9$$: Choose a test point such as $$x = 5$$. Then, $$(5 - 4)(5 - 9) = (1)(-4) = -4 < 0$$. So the expression is negative. - For $$x > 9$$: Choose a test point such as $$x = 10$$. Then, $$(10 - 4)(10 - 9) = (6)(1) = 6 > 0$$. So the expression is positive. **Step 5. Write the solution** Since we want the expression to be greater than zero, we take the intervals where the product is positive: $$ x < 4 \quad \text{or} \quad x > 9. $$ **Final Answer** - In inequality notation: $$x < 4$$ or $$x > 9$$ - In interval notation: $$(-\infty, 4) \cup (9, \infty)$$ The solution in interval notation is $$ (-\infty, 4) \cup (9, \infty) $$.

Use a sign chart to solve the inequality. Express the answer in inequality and interval notation.

The solution expressed in inequality notation is or .
(Type an inequality or a compound inequality.)
The solution expressed in interval notation is

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Algebra Questions

Use a sign chart to solve the inequality. Express the answer in inequality and interval notation. The solution expressed in inequality notation is or . (Type an inequality or a compound inequality.) The solution expressed in interval notation is

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Algebra Questions

Use a sign chart to solve the inequality. Express the answer in inequality and interval notation.

The solution expressed in inequality notation is or .
(Type an inequality or a compound inequality.)
The solution expressed in interval notation is