Solving Basic Quadratic Inequalities Easily

Solving basic quadratic inequalities does not have to be hard. With a few simple steps, anyone can learn how to solve them. These inequalities involve expressions where the variable is squared. They are very common in math problems, physics, and business. In this guide, we will break down each step so you can solve it easily and confidently.

What Is a Quadratic Inequality?

A quadratic inequality looks like a quadratic equation but uses an inequality sign instead of an equal sign. It can look like this:

• x2+3x−10>0x^2 + 3x – 10 > 0x2+3x−10>0

• x2−2x+1≤0x^2 – 2x + 1 \leq 0x2−2x+1≤0
  The variable is squared, and we want to find which values make the inequality true.

Step 1: Write the Inequality in Standard Form

First, write the inequality in standard form. Move all terms to one side so the other side is zero. For example, if the inequality is x2<5x+6x^2 < 5x + 6x2<5x+6, subtract $5x+6$ from both sides to get:
x2−5x−6<0x^2 – 5x – 6 < 0x2−5x−6<0
Now you are ready to solve it.

Step 2: Solve the Related Equation

Replace the inequality sign with an equal sign.
So now solve:
x2−5x−6=0x^2 – 5x – 6 = 0x2−5x−6=0
Factor it:
$(x-6)(x+1)=0$
Now find the roots:
$x=6$ and $x=-1$
These are the points where the expression equals zero. They will help us find the solution to the inequality.

Step 3: Make Intervals on the Number Line

The two roots divide the number line into three parts:

• $(-\infty ,-1)$

• $(-1,6)$

• $(6,\infty )$
  We will test one number from each interval to see where the inequality is true.

Step 4: Test Each Interval

Choose a number from each interval and plug it into the original inequality x2−5x−6<0x^2 – 5x – 6 < 0x2−5x−6<0

• Test $x=-2$:
  (−2)2−5(−2)−6=4+10−6=8(-2)^2 – 5(-2) – 6 = 4 + 10 – 6 = 8(−2)2−5(−2)−6=4+10−6=8
  Is $8<0$? No

• Test $x=0$:
  $0-0-6=-6$
  Is $-6<0$? Yes

• Test $x=7$:
  $49-35-6=8$
  Is $8<0$? No
  So only the interval $(-1,6)$ makes the inequality true.

Step 5: Write the Final Answer

Now write the solution. Since the inequality sign is strict ($<$), we use parentheses. The final answer is:
$(-1,6)$

Solving Inequalities With $\ge$ or $\le$

Let’s look at another example:
Solve x2−4x+3≤0x^2 – 4x + 3 \leq 0x2−4x+3≤0
Step 1: It is already in standard form
Step 2: Solve the equation x2−4x+3=0x^2 – 4x + 3 = 0x2−4x+3=0
Factor: $(x-3)(x-1)=0$
Roots: $x=1$ and $x=3$
Step 3: Intervals are: $(-\infty ,1)$, $(1,3)$, $(3,\infty )$
Step 4: Test a number from each:

• $x=0$: $0-0+3=3$, not less than or equal to zero

• $x=2$: $4-8+3=-1$, yes

• $x=4$: $16-16+3=3$, no
  Step 5: The inequality is true between 1 and 3, and we include the endpoints because of the $\le$ sign
  Final answer: $[1,3]$

Important Tips

• Always write the inequality so that one side is zero

• Solve the related equation to find the boundary points

• Use the boundary points to split the number line into intervals

• Test each interval using a simple value from that section

• Choose brackets $[]$ for $\le$ or $\ge$, and parentheses $()$ for $<$ or $>$

• Never skip testing intervals. This is the key step in finding the right solution

Why This Matters

Quadratic inequalities appear in many real problems. You might use them to figure out where a business makes profit, where a ball stays above a certain height, or where costs remain under control. Knowing how to solve them makes you better at math and problem-solving.

Conclusion

Solving basic quadratic inequalities easily is all about following the right steps. First, get the inequality in the correct form. Then solve the related equation. Use the roots to divide the number line. Test values and write the solution using interval notation. With practice, you can solve any basic quadratic inequality without stress.