Solve the quadratic inequality.
$3 x(x-1) \leq 10-2 x$
Step 1: Move all terms to one side
Step 2: Make it an equation, solve for $x$ (find the zeros)
Answer (1)
Expand and rearrange the inequality: 3 x 2 − x − 10 ≤ 0 .
Factor the quadratic equation: ( 3 x + 5 ) ( x − 2 ) = 0 , find the zeros x = − 3 5 and x = 2 .
Determine the interval where the inequality holds: − 3 5 ≤ x ≤ 2 .
Express the solution in interval notation: x ∈ [ − 3 5 , 2 ] .
Explanation
Rearrange the Inequality Let's solve the quadratic inequality step by step! First, we need to rearrange the inequality so that all the terms are on one side, making it easier to work with.
Simplify the Inequality We start with the given inequality: 3 x ( x − 1 ) ≤ 10 − 2 x Expanding the left side, we get: 3 x 2 − 3 x ≤ 10 − 2 x Now, we move all the terms to the left side: 3 x 2 − 3 x + 2 x − 10 ≤ 0 Simplifying, we have: 3 x 2 − x − 10 ≤ 0
Find the Zeros Now that we have a standard quadratic inequality, we need to find the zeros of the corresponding quadratic equation. This will help us determine the intervals where the inequality holds true.
Solve the Quadratic Equation We consider the quadratic equation: 3 x 2 − x − 10 = 0 We can solve this equation by factoring. We look for two numbers that multiply to 3 c d o t − 10 = − 30 and add up to − 1 . These numbers are − 6 and 5 . So we can rewrite the middle term as − 6 x + 5 x : 3 x 2 − 6 x + 5 x − 10 = 0 Now, we factor by grouping: 3 x ( x − 2 ) + 5 ( x − 2 ) = 0 ( 3 x + 5 ) ( x − 2 ) = 0 This gives us two possible solutions for x : 3 x + 5 = 0 R i g h t a rro w x = − 3 5 x − 2 = 0 R i g h t a rro w x = 2 So the zeros of the quadratic equation are x = − 3 5 and x = 2 .
Determine the Intervals Now that we have the zeros, we can determine the intervals where the inequality 3 x 2 − x − 10 ≤ 0 is satisfied. Since the coefficient of x 2 is positive (3), the parabola opens upwards. This means that the quadratic expression is negative between the roots.
Write the Solution Therefore, the solution to the inequality is the interval between the roots, including the roots themselves: − 3 5 ≤ x ≤ 2 In interval notation, this is: x ∈ [ − 3 5 , 2 ]
Final Answer So, the solution to the quadratic inequality 3 x ( x − 1 ) ≤ 10 − 2 x is x ∈ [ − 3 5 , 2 ]
Examples
Understanding quadratic inequalities is very useful in various real-world scenarios. For example, imagine a company wants to determine the range of production levels that will ensure their profit stays above a certain threshold. If the profit function is modeled by a quadratic equation, solving a quadratic inequality will help them find the minimum and maximum production levels needed to achieve their desired profit. Another example is in physics, where projectile motion can be described by quadratic equations. Solving quadratic inequalities can help determine the time interval during which the projectile is above a certain height.
