3Material AICLE. 4º de ESO: Equations and Inequalities (Solucionario)

EQUATIONSAND

INEQUALITIES

- Key -

4 Material AICLE. 4º de ESO: Equations and Inequalities (Solucionario)

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Linear Equation 52=

−xx  

Quadratic Equation

Biquadratic Equation

Equation containing Algebraic Fractions

Radical Equation

Inequality

0122 =+− xx  

2x + 3 = 5(x −1)

912 ≥+− x  

52 −= xx

086 24 =+− xx  

key words sign of inequality

is greater thanis more than

has more

is less thanis fewer than

has fewer

is greater than or equal tois at least

has at least

is less than or equal tois at most

has at most

>

<

≥

≤

coefficient variable operator constants expression

terms

coefficient

expression

variable operator constants

terms

5Material AICLE. 4º de ESO: Equations and Inequalities (Solucionario)

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8 + x < 12 eight plus a number is less than twelve.

2 x = 20 two times a number equals twenty.

x / 2 > 9 half a number is less than nine.

2 x + 5 = 8 two times a number plus five equals eight.

- 2 x = 152 minus two times a number equals one hundred and fifty two.

x / 3 + 1= 67 one third of a number plus one equals sixty seven.

21 x + 5 ≥ - 7 twenty times a number plus five is at least minus seven.

x2 – 9 x + 1 > 0 a number squared minus nine times the same number plus one is greater than zero.

Steps for solving linear equations

1. Clear parentheses (by distributive property)2. Clear fractions (multiply each term in the equation by the LCD)3. Combine similar terms on each side of the equal sign.4. Use addition/subtraction to put all variable terms on one side of the equation and all constants on the other side.5. Divide both sides of the equation by the coefficient of x.6. Simplify your answer.

Steps for solving a quadratic equation using the quadratic formula

1. Write the equation in standard form, ax2+ bx + c = 0.2. Determine the values of a, b, and c; a is the coefficient of x, b is the coefficient of x, and c is the constant.3. Substitute these values of a, b, and c into the quadratic formula.4. Simplify.

6 Material AICLE. 4º de ESO: Equations and Inequalities (Solucionario)

Steps for solving a biquadratic equation

1. Simplify the equation and write it in standard form, ax4+ bx2 + c = 0

2. Call x2 = z , which means that x4 = z2

3. Solve the equation with the new unknown factor (apply the quadratic formula)

4. Use these values of z to work out the values of x: z = x2, from which: x = the square root of z

Steps for solving an equation containing algebraic fractions

1. Find any excluded values (values of the variable that would make the denominator 0).2. Find the LCD of the entire equation.3. Multiply both sides of the equation by the LCD.4. Solve the equation remaining.5. Check for extraneous solutions.

Steps for solving inequalities

1. Solve the inequality as though it were an equation. The real solutions to the equation become boundary points for the solution to the inequality.

2. Make the boundary points solid circles if the original inequality includes equality; otherwise, make the boundary points open circles.

3. Select points from each of the regions created by the boundary points. Replace these “test points” in the original inequality.

4. If a test point satisfies the original inequality, then the region that contains that test point is part of the solutions.

5. Represent the solution in graphic form and in solution set form

Steps for solving radical equations

1. Isolate one of the radicals (get one radical on one side and everything else on the other using inverse operations)2. Get rid of your radical sign (the inverse operation to a radical or a root is to raise it to an exponent)3. If you still have a radical sign left, repeat steps 1 and 2.4. Solve the remaining equation.5. Check for extraneous solutions.

7Material AICLE. 4º de ESO: Equations and Inequalities (Solucionario)

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15.

b) (A) x = -10 is not a solution (B) x = -10 is a solution

c) (A) x = 3 (B) x = -10

a) (A) two solutions (B) two solutions (C) two solutions

b) (A) x = 1 and x = -1/3 (B) x = ± 12 / 7 (C) x = 0 and x = -3/2

a) (A) biquadratic (B) containing algebraic fractions (C) radical equation

b) (A) x = ± 5 and x = ±2 (B) x = 1 and x = -10/3 (C) x = 5 and x = 20/9

Read key variable equation add subtract times Check

a) The number is 15.b) t equals 3 years.c) The return trip took him 25 minutes.d) The length and width are 8 feet and 3 feet.e) The must sell at least 182 buttons.

(A) x ≤ -9 (B) -3 ≤ x ≤ 3 (C) x < -2 and x ≥ 1

a) x = -5 and x = 3c) x = -2 and x = -2/3b) x = 1/9 and x = 25/4d) x = -5 and x = -1