Solve Quadratic Inequality: x²-4x+3≤(3x+5)(2x-3)

In summary, a quadratic inequality is an inequality that involves a quadratic expression and its solution is a range of values for the variable. To solve a quadratic inequality, you need to move terms to one side, factor the expression, find critical values, plot them on a number line, and write the solution in interval notation. Solving a quadratic equation gives a specific value while solving a quadratic inequality gives a range of values. The quadratic formula can be used to solve a quadratic inequality by rearranging the inequality and finding critical values. To graph a quadratic inequality, rewrite it in standard form, find the vertex, plot points, draw a parabola, use a dashed or solid line and shade the correct region.
  • #1
moondaaay
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0

Homework Statement



How to solve this kind of inequality?
x²-4x+3≤(3x+5)(2x-3)

Homework Equations

The Attempt at a Solution

:[/B]
I'm confused. Should I factor the left side or should I FOIL the right side then equate it to zero to find the critical numbers? Help pleaasee.
 
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  • #2
Simplify the inequality and arrange for all terms to be on one side and 0 on the other side. The quadratic inequality can then be examined for the three intervals on x. The critical values will be the roots of the simplified quadratic expression.
 

FAQ: Solve Quadratic Inequality: x²-4x+3≤(3x+5)(2x-3)

What is a quadratic inequality?

A quadratic inequality is an inequality that involves a quadratic expression, which is an expression in the form of ax^2 + bx + c, where a, b, and c are constants and x is a variable. The solution to a quadratic inequality is a range of values for the variable that satisfies the inequality.

How do you solve a quadratic inequality?

To solve a quadratic inequality, you need to follow these steps:

  1. Move all terms to one side of the inequality so that the other side is equal to 0.
  2. Factor the quadratic expression, if possible.
  3. Determine the critical values by setting each factor equal to 0 and solving for the variable.
  4. Plot the critical values on a number line and determine which intervals satisfy the inequality.
  5. Write the solution in interval notation.

What is the difference between solving a quadratic equation and a quadratic inequality?

A quadratic equation has an equal sign, while a quadratic inequality has an inequality sign. This means that solving a quadratic equation will give you a specific value for the variable, while solving a quadratic inequality will give you a range of values that satisfy the inequality.

Can you use the quadratic formula to solve a quadratic inequality?

Yes, you can use the quadratic formula to solve a quadratic inequality. However, you will need to rearrange the inequality so that one side is equal to 0 before using the formula. The solutions obtained from the quadratic formula will be the critical values, which can then be used to determine the intervals that satisfy the inequality.

How do you graph a quadratic inequality?

To graph a quadratic inequality, you will need to follow these steps:

  1. Rewrite the inequality in standard form, with one side equal to 0.
  2. Identify the vertex of the parabola by finding the x-coordinate using -b/2a, where a and b are the coefficients of x^2 and x, respectively.
  3. Plot the vertex on the coordinate plane.
  4. Find two additional points by substituting values for x and solving for y.
  5. Plot these points and draw a parabola through them.
  6. Use a dashed or solid line to indicate whether the inequality is less than or equal to, or greater than or equal to.
  7. Shade the region that satisfies the inequality.

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