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- 1. Solving Equations and Inequalities Lindsay Lehman And Dana Bailey
- 2. Linear Equations <ul><li>5x + 3 = 28 </li></ul><ul><li>-3 -3 </li></ul><ul><li>5x = 25 </li></ul><ul><li>÷ 5 ÷5 </li></ul><ul><li>x = 5 </li></ul>
- 3. Quadratic Equations <ul><li>There are multiple ways to solve quadratic equations. These include: </li></ul><ul><ul><li>Factoring </li></ul></ul><ul><ul><li>The quadratic formula </li></ul></ul><ul><ul><li>Completing the square </li></ul></ul>
- 4. Factoring <ul><li>x^2 + 8x = -15 </li></ul><ul><li>+15 +15 </li></ul><ul><li>x^2 + 8x + 15 = 0 </li></ul><ul><li>(x + 3)(x + 5) = 0 </li></ul><ul><li>x + 3 = 0 x + 5 = 0 </li></ul><ul><li>x = -3 and x = -5 </li></ul><ul><li>Always plug your solutions back into the original equation to check for extraneous solutions. </li></ul>
- 5. Using the Quadratic Formula <ul><li>The quadratic formula is: </li></ul><ul><ul><ul><li>(-b) ± √(b^2)-4(a)(c) </li></ul></ul></ul><ul><ul><ul><li>2(a) </li></ul></ul></ul><ul><li>When trying to solve quadratic equations and factoring doesn’t work, the alternative is to use the quadratic formula. It always works. </li></ul><ul><li>Simply take the parts of the equation and plug them into the formula. Then solve. </li></ul>
- 6. Quadratic Formula Example <ul><li>x^2 + 4x – 7 = 0 </li></ul><ul><li>This equation does not factor, or at least not easily. Simply identify a, b, and c and plug them in to the quadratic formula. </li></ul><ul><li>For this equation: </li></ul><ul><ul><li>A = 1 </li></ul></ul><ul><ul><li>B = 4 </li></ul></ul><ul><ul><li>C = -7 </li></ul></ul>
- 7. Quadratic Example Continued <ul><li>X = (-4)± √4^2 – 4(1)(-7) </li></ul><ul><li>2(1) </li></ul><ul><li>Now just solve </li></ul><ul><li>(-4) + 2√11 = X (-4) – 2√11 = X </li></ul><ul><li>2 2 </li></ul>
- 8. Completing the Square <ul><li>Another alternative to use when a quadratic equation may not be factored easily is completing the square. </li></ul><ul><li>When you have an equation: </li></ul><ul><ul><li>Ax^2 + Bx + C = 0 </li></ul></ul><ul><li>Just pull C to the opposite side (by subtracting or adding it). </li></ul><ul><li>Then, to make Ax^2 + Bx a perfect square, just add (b/2)^2 to both sides of the equation. </li></ul><ul><li>Once this is completed, write the equation in factored form, (x + b/2)^2 = c + (b/2)^2, then solve. </li></ul>
- 9. Completing the Square Example <ul><li>x^2 + 2x - 4 = 0 </li></ul><ul><li>x^2 + 2x + __ = 4 </li></ul><ul><li>x^2 + 2x + 1 = 4 + 1 </li></ul><ul><li>(x + 1)^2 = 5 </li></ul><ul><li>x + 1 = ± √5 </li></ul><ul><li>x = -1 ± √5 </li></ul>
- 10. Rational Equations <ul><li>Rational equations are basically fractions set equal to each other with variables in them. </li></ul><ul><li>All you have to do is get a common denominator (so that you can cancel them out), then solve the two expressions set equal to each other. </li></ul><ul><li>Or, if there is only a denominator on one side, you can multiply both sides by the common denominator to get rid of it, then solve. </li></ul>
- 11. Rational Equation Example <ul><li>2x + 5 = 15 </li></ul><ul><li>4 </li></ul><ul><li>2x + 5 = (15x)4 </li></ul><ul><li>2x + 5 = 60 </li></ul><ul><li>2x = 55 </li></ul><ul><li>x = 27.5 </li></ul>
- 12. Radical Equations <ul><li>Radical Equations are equations that have radicals in them. They look tricky to deal with, but when you know how to handle them it’s not so bad. </li></ul>
- 13. How to Solve Radical Equations <ul><li>Solving radical is actually very simple. </li></ul><ul><li>Just follow these steps: </li></ul><ul><li>First, one must get the radical by itself on one side of the equation. </li></ul><ul><li>Then, one must square both sides (in order to eliminate the square root). </li></ul><ul><li>Once the square root is gone, simply solve the equation. </li></ul><ul><li>Finally, NEVER forget to check all solutions to weed out any extraneous ones. </li></ul>
- 14. Radical Equation Example <ul><li>x + 8 = √(5 + 2x) + 12 </li></ul><ul><li>x + 8 = √(5 + 2x) + 12 </li></ul><ul><li>-12 -12 </li></ul><ul><li>(x – 4)^2 = (√(5 + 2x))^2 </li></ul><ul><li>x^2 – 8x + 16 = 5 +2x </li></ul><ul><li>-(5 + 2x) -(5 + 2x) </li></ul><ul><li>x^2 – 10x + 11 = 0 </li></ul>
- 15. Radical Equation Ex. Continued <ul><li>x = -(-10) ± √( 100 – 4(1)(11)) </li></ul><ul><li>2(1) </li></ul><ul><li>x = 10 ± √(56) </li></ul><ul><li>2 2 </li></ul><ul><li>= 5 ± √(14) </li></ul><ul><li>Solutions: </li></ul><ul><li>x ≈ 8.74, x ≈ 1.26 </li></ul>
- 16. Inequalities <ul><li>Inequalities are equations with symbols in place of where the equals sign would be in a regular equation. There are four different signs, they are: </li></ul><ul><ul><ul><li>> - Greater than </li></ul></ul></ul><ul><ul><ul><li>< - Less than </li></ul></ul></ul><ul><ul><ul><li>≤ - Less than or equal to </li></ul></ul></ul><ul><ul><ul><li>≥ - Greater than or equal to </li></ul></ul></ul>
- 17. Linear Inequalities <ul><li>Solving a linear inequality is basically like solving a linear equation. The only difference is that when you divide or multiply by a negative, you have to flip the symbol around. </li></ul><ul><li>For example: </li></ul><ul><li>-2x + 5 > 15 </li></ul><ul><li>-2x > 10 </li></ul><ul><li>x < 5 </li></ul>
- 18. Quadratic Inequalities <ul><li>Quadratic inequalities are basically solved the same way quadratic equations are solved. </li></ul><ul><li>For example: </li></ul><ul><li>x^2 + 5x + 6 > 0 </li></ul><ul><li>(x + 2)(x + 3) > 0 </li></ul><ul><li>x + 2 > 0 x + 3 > 0 </li></ul><ul><li>Solutions: </li></ul><ul><li>x > -2 x > -3 </li></ul>
- 19. Solving by Graphing <ul><li>Another way to solve any type of equation is to graph it. Generally, the easiest way to graph an equation/function is to first make a table. Then, just plot the points. </li></ul>
- 20. Solving by Graphing Continued <ul><li>Linear equations: </li></ul><ul><ul><ul><li>Just make a table to gather a few points, then graph them. It should look like a straight line. From the graph you can find both the x and y intercepts. </li></ul></ul></ul><ul><li>Quadratic equations: </li></ul><ul><ul><ul><li>Just like for linear equations, make a table and plot a few points. It would also be beneficial to find the vertex. By looking at the graph you can find the x and y intercepts and if it is a maximum or a minimum. Solving a quadratic equation by graphing means looking for the zeros. </li></ul></ul></ul>
- 21. Solving by Graphing Continued <ul><li>Radical Equations: </li></ul><ul><ul><ul><li>You can either make a table to gather several values or you can go from what you know about radical functions and just graph from there. (ie- √(x+5) just move to the left 5, and start curve on x-axis). </li></ul></ul></ul><ul><li>Rational Equations: </li></ul><ul><ul><ul><li>Again, you can either make a table or go off of what you already know about rational equations. </li></ul></ul></ul>
- 22. Solving by Graphing Continued <ul><li>Linear and Quadratic Inequalities: </li></ul><ul><ul><ul><li>First make a table and graph the line/parabola as you would for a linear/quadratic equation. To decide where to shade plug in the point (0, 0). If the point works in the equation then shade on the side of the line/parabola that the point is on. If it does not work when you plug it in, shade on the opposite side where that point is not. </li></ul></ul></ul>

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