Republic of the Philippines
DEPARTMENT OF EDUCATION
Region IX, Zamboanga Peninsula
SCHOOL DIVISION OF ZAMBAONGA DEL NORTE
Siocon 2 District, Siocon, Zamboanga Del Norte
LEARNING ACTIVITY SHEETS
MATHEMATICS 9 QUARTER 1, WEEK 5
Name the Learner:______________Grade Level &Section:____________
School:____________________________Date:_________Score:____________
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Learning Competency: illustrate quadratic inequalities, solves
quadratic inequalities, solves problem involving quadratic
inequalities. ( Q1-Week 5, M9AL-If-1.M9AL-If-2,M9AL-If-g-1)
Learning Objectives: At the of the lesson student are able to
illustrate and solve problems involving quadratic inequalities.
Tittle/Topic: Quadratic Inequalities
References: Mathematics – Grade 9 Learner’s Material Mathematics
Grade 9
Concept Notes:
A. What to know?
this lesson assessing your knowledge of the different mathematics
concepts previously and your skills in performing mathematical
operations. These knowledge and skills will help you in
understanding quadratic inequalities. As you go through this lesson ,
think of this important question: ”how are quadratic inequalities used
in solving real-life problems and making decisions?’
B. Definition of Terms:
Quadratic inequality- is an inequality that contains a polynomial
of degree 2 and can be written in any of the following forms.
Quadratic inequality- is an equation of second degree that uses an
inequality sign instead of an equal sign.
solutions to quadratic inequality always give the two roots. The nature of
the roots may differ and can be determined by discriminant (b 2 – 4ac).
C. Activities:
The General forms of the quadratic inequalities are:
ax²+bx+c>0
ax²+bx+c>0
ax²+bx+c<0
ax²+bx+c<0
where a, b, and c are real numbers and a≠0.
How to Solve Quadratic Inequalities?
A quadratic inequality is an equation of second degree that uses an
inequality sign instead of an equal sign.
Examples of quadratic inequalities
x2 – 6x – 16 ≤ 0,
x2 + 4 > 0,
x2
2x2 – 11x + 12 > 0,
– 3x + 2 ≤ 0 etc.
Solving a quadratic inequality in Algebra is similar to solving a
quadratic equation. The only exception is that, with quadratic
equations, you equate the expressions to zero, but with inequalities,
you’re interested in knowing what’s on either side of the zero i.e.
negatives and positives.
Quadratic equations can be solved by either the factorization
method or by use of the quadratic formula.
Before we can learn how to solve quadratic inequalities, let’s
recall how quadratic equations are solved by handling a few
examples.
How Quadratic Equations are Solved by Factorization Method?
Since we know we can similarly solve quadratic inequalities as quadratic
equations, it is useful to understand how to factorize the given equation or
inequality.
Let’s see a few examples here.
1. Solve 6x2– 7x + 2 = 0
Solution
⟹ 6x2 – 4x – 3x + 2 = 0
Factorize the expression;
⟹ 2x (3x – 2) – 1(3x – 2) = 0
⟹ (3x – 2) (2x – 1) = 0
⟹ 3x – 2 = 0 or 2x – 1 = 0
⟹ 3x = 2 or 2x = 1
⟹ x = 2/3 or x = 1/2
Therefore, x = 2/3,
2. Solve 3x2– 6x + 4x – 8 = 0
Solution
Factorize the expression on the left-hand side.
⟹ 3x2 – 6x + 4x – 8 = 0
⟹ 3x (x – 2) + 4(x – 2) = 0
⟹ (x – 2) (3x + 4) = 0
⟹ x – 2 = 0 or 3x + 4 = 0
⟹ x = 2 or x = -4/3
Therefore, the roots of the quadratic equation are, x = 2, -4/3.
3. Expand the equation;
x2 + 4x – 3xy – 12y = 0
Factorize;
⟹ x (x + 4) – 3y (x + 4) = 0
x + 4) (x – 3y) = 0
⟹ x + 4 = 0 or x – 3y = 0
⟹ x = -4 or x = 3y
Thus, x = -4 or x = 3y
To solve a quadratic inequality, we also apply the same method as illustrated in the
procedure below:
1.Write the quadratic inequality in standard form:
ax2 + bx + c where a, b and are coefficients and a ≠ 0
2. Determine the roots of the inequality.
3. Write the solution in inequality notation or interval notation.
4. If the quadratic inequality is in the form:(x – a) (x – b) ≥ 0, then a ≤ x ≤ b,
and if it is in the form :(x – a) (x – b) ≤ 0, when a < b then a ≤ x or x ≥ b.
Example 1.Solve the inequality x2 – 4x > –3
Solution
First, make one side one side of the inequality zero by adding both sides by
3.
x2 – 4x > –3 ⟹ x2 – 4x + 3 > 0
Factor the left side of the inequality.
x2 – 4x + 3 > 0 ⟹ (x – 3) (x – 1) > 0
Solve for all the zeroes for the inequality;
For, (x – 1) > 0 ⟹ x > 1 and for, (x – 3) > 0 ⟹ x>3
Since y is positive, we therefore choose the values of x which the curve will
be above the x-axis.
x < 1 or x > 3
Example 2. Solve the inequality x2 – x > 12.
Solution
To write the inequality in standard form, subtract both sides of the
inequality by 12.
x2 – x > 12 ⟹ x2 – x – 12 > 0.
Factorize the quadratic inequality to get to;
(x – 4) (x + 3) > 0
Solve for all the zeroes for the inequality;
For, (x + 3) > 0 ⟹ x > -3
For x – 4 > 0 ⟹ x > 4
The values x < –3 or x > 4 are therefore the solution of this quadratic
inequality.
Example 3. Solve 2x2 < 9x + 5
Solution:
Write the inequality in standard form by making one side of the inequality zero.
2x2 < 9x + 5 ⟹ 2x2 – 9x – 5 < 0
Factor the left side of the quadratic inequality.
2x2 – 9x – 5 < 0 ⟹ (2x + 1) (x – 5) < 0
Solve for all the zeroes for the inequality
For, (x – 5) < 0 ⟹ x < 5 and for (2x + 1) < 0 ⟹ x < -1/2
Since y is negative for the equation 2x2 – 9x – 5 < 0,
we therefore choose the values of x which the curve will be below the x axis.
Therefore, the solution is -1/2 < x < 5
Weekly Assessment!
Direction: Solve the following ,incircle the letter of the correct answer.
1.(x − 3) (x + 1) < 0
a. −1 < x < 3
c. x < −4/3 or x > ½
2
2. x + 5x + 6 ≥ 0
a. −1 < x < 3
c. x < −4/3 or x > ½
3. (2x − 1) (3x + 4) > 0
a. −1 < x < 3
c. x < −4/3 or x > ½
2
4. 10x − 19x + 6 ≤ 0
a. −1 < x < 3
c. x < −4/3 or x > ½
5. 5 − 4x − x 2 > 0
a. −5 < x < 1
c. x < −4/3 or x > ½
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b. x < −3 or x > −2
d. 2/5 ≤ x ≤ 3/2\
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b. x < −3 or x > −2
d. 2/5 ≤ x ≤ 3/2\
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b. x < −3 or x > −2
d. 2/5 ≤ x ≤ 3/2\
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b. x < −3 or x > −2
d. 2/5 ≤ x ≤ 3/2\
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b. x < −3 or x > −2
d. 2/5 ≤ x ≤ 3/2\
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Good luck & keep safe
Prepared by:
CORY S. IGNACIO
Note” please do show your solution on the space provided.