Answer in Interval Notation

Oct 13, 2025 by windailys.com 12 views

Inequalities are mathematical expressions that compare two values, often using greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) symbols. Solving inequalities involves isolating the variable on one side of the inequality sign and finding the values that satisfy the inequality. In this article, we will explore four different types of inequalities and provide step-by-step solutions using interval notation.

Inequality 1: Linear Inequality

Problem Statement

Solve the inequality: $4 - 3x \leq 7 + 2x$

Solution

To solve this inequality, we need to isolate the variable x on one side of the inequality sign.

4 - 3x ≤ 7 + 2x
-3x - 2x ≤ 7 - 4
-5x ≤ 3
x ≥ -3/5

Explanation

We started by subtracting 2x from both sides of the inequality to get -5x on the left side. Then, we subtracted 3 from both sides to get 3 on the right side. Finally, we divided both sides by -5 to isolate x. Since we divided by a negative number, we flipped the direction of the inequality sign.

Answer in Interval Notation

The solution to this inequality is $x \in \left[-\frac{3}{5}, \infty\right)$ .

Inequality 2: Compound Inequality

Problem Statement

Solve the inequality: $-2 \ \textless \ 5 + 3x \leq 20$

Solution

To solve this compound inequality, we need to break it down into two separate inequalities.

-2 < 5 + 3x
-7 < 3x
x > -7/3
5 + 3x ≤ 20
3x ≤ 15
x ≤ 5

Explanation

We started by subtracting 5 from both sides of the first inequality to get -7 on the left side. Then, we divided both sides by 3 to isolate x. Since we divided by a positive number, the direction of the inequality sign remains the same. For the second inequality, we subtracted 3x from both sides to get 3x on the left side. Then, we subtracted 15 from both sides to get 0 on the right side. Finally, we divided both sides by 3 to isolate x.

Answer in Interval Notation

The solution to this compound inequality is $x \in \left(-\frac{7}{3}, 5\right]$ .

Inequality 3: Absolute Value Inequality

Problem Statement

Solve the inequality: $|2 - 7x| - 1 \ \textgreater \ 4$

Solution

To solve this absolute value inequality, we need to isolate the absolute value expression on one side of the inequality sign.

|2 - 7x| - 1 > 4
|2 - 7x| > 5
2 - 7x > 5 or 2 - 7x < -5
-7x > 3 or -7x < -7
x < -3/7 or x > 1

Explanation

We started by adding 1 to both sides of the inequality to get 5 on the right side. Then, we removed the absolute value sign by creating two separate inequalities. For the first inequality, we subtracted 2 from both sides to get -7x on the left side. Then, we divided both sides by -7 to isolate x. Since we divided by a negative number, we flipped the direction of the inequality sign. For the second inequality, we subtracted 2 from both sides to get -7x on the left side. Then, we divided both sides by -7 to isolate x. Since we divided by a negative number, we flipped the direction of the inequality sign.

Answer in Interval Notation

The solution to this absolute value inequality is $x \in \left(-\infty, -\frac{3}{7}\right) \cup \left(1, \infty\right)$ .

Inequality 4: Absolute Value Inequality

Problem Statement

Solve the inequality: $|5 - 3x| \leq 12$

Solution

To solve this absolute value inequality, we need to isolate the absolute value expression on one side of the inequality sign.

|5 - 3x| ≤ 12
5 - 3x ≤ 12 or 5 - 3x ≥ -12
-3x ≤ 7 or -3x ≥ -17
x ≥ -7/3 or x ≤ 17/3

Explanation

We started by removing the absolute value sign by creating two separate inequalities. For the first inequality, we subtracted 5 from both sides to get -3x on the left side. Then, we divided both sides by -3 to isolate x. Since we divided by a negative number, we flipped the direction of the inequality sign. For the second inequality, we subtracted 5 from both sides to get -3x on the left side. Then, we divided both sides by -3 to isolate x. Since we divided by a negative number, we flipped the direction of the inequality sign.

Answer in Interval Notation

The solution to this absolute value inequality is $x \in \left[-\frac{17}{3}, -\frac{7}{3}\right] \cup \left[0, \frac{17}{3}\right]$ .

In the previous article, we explored four different types of inequalities and provided step-by-step solutions using interval notation. In this article, we will answer some frequently asked questions about solving inequalities.

Q: What is the difference between a linear inequality and a compound inequality?

A: A linear inequality is an inequality that can be written in the form ax + b ≤ c or ax + b ≥ c, where a, b, and c are constants. A compound inequality is an inequality that involves two or more inequalities joined by the words "and" or "or". For example, the inequality 2 < x ≤ 5 is a compound inequality.

Q: How do I solve an absolute value inequality?

A: To solve an absolute value inequality, you need to isolate the absolute value expression on one side of the inequality sign. Then, you can remove the absolute value sign by creating two separate inequalities. For example, the inequality |x| ≤ 3 can be solved by creating two separate inequalities: x ≤ 3 and x ≥ -3.

Q: What is interval notation?

A: Interval notation is a way of writing the solution to an inequality using a specific notation. For example, the solution to the inequality x ≥ 2 can be written in interval notation as [2, ∞). The square brackets indicate that the value 2 is included in the solution, and the infinity sign indicates that the solution extends to infinity.

Q: How do I determine the direction of the inequality sign when solving an inequality?

A: When solving an inequality, you need to determine the direction of the inequality sign based on the operation you are performing. For example, if you are multiplying both sides of the inequality by a negative number, you need to flip the direction of the inequality sign.

Q: Can I use a calculator to solve inequalities?

A: Yes, you can use a calculator to solve inequalities. However, you need to be careful when using a calculator to solve inequalities, as it may not always give you the correct solution. It's always a good idea to check your solution by plugging it back into the original inequality.

Q: How do I graph the solution to an inequality on a number line?

A: To graph the solution to an inequality on a number line, you need to plot a point on the number line that represents the solution to the inequality. Then, you need to shade the region on the number line that represents the solution. For example, the solution to the inequality x ≥ 2 can be graphed on a number line by plotting a point at 2 and shading the region to the right of 2.

Q: Can I solve inequalities with fractions?

A: Yes, you can solve inequalities with fractions. However, you need to be careful when working with fractions, as they can be tricky to work with. It's always a good idea to simplify fractions before solving the inequality.

Q: How do I solve inequalities with exponents?

A: To solve inequalities with exponents, you need to isolate the exponent on one side of the inequality sign. Then, you can use the properties of exponents to simplify the inequality. For example, the inequality 2^x ≤ 8 can be solved by isolating the exponent x and then using the properties of exponents to simplify the inequality.

Q: Can I solve inequalities with radicals?

A: Yes, you can solve inequalities with radicals. However, you need to be careful when working with radicals, as they can be tricky to work with. It's always a good idea to simplify radicals before solving the inequality.

Q: How do I determine the solution to an inequality with multiple variables?

A: To determine the solution to an inequality with multiple variables, you need to isolate one variable on one side of the inequality sign. Then, you can use the properties of inequalities to simplify the inequality. For example, the inequality x + y ≤ 5 can be solved by isolating one variable, say x, and then using the properties of inequalities to simplify the inequality.

Q: Can I use algebraic methods to solve inequalities?

A: Yes, you can use algebraic methods to solve inequalities. Algebraic methods involve using algebraic properties and operations to simplify the inequality and isolate the variable. For example, the inequality x^2 + 4x + 4 ≤ 0 can be solved using algebraic methods by factoring the quadratic expression and then using the properties of inequalities to simplify the inequality.

Q: How do I determine the solution to an inequality with absolute value?

A: To determine the solution to an inequality with absolute value, you need to isolate the absolute value expression on one side of the inequality sign. Then, you can remove the absolute value sign by creating two separate inequalities. For example, the inequality |x| ≤ 3 can be solved by creating two separate inequalities: x ≤ 3 and x ≥ -3.

Q: Can I use graphical methods to solve inequalities?

A: Yes, you can use graphical methods to solve inequalities. Graphical methods involve graphing the solution to the inequality on a number line or a coordinate plane. For example, the solution to the inequality x ≥ 2 can be graphed on a number line by plotting a point at 2 and shading the region to the right of 2.

Q: How do I determine the solution to an inequality with multiple absolute values?

A: To determine the solution to an inequality with multiple absolute values, you need to isolate each absolute value expression on one side of the inequality sign. Then, you can remove each absolute value sign by creating two separate inequalities. For example, the inequality |x| + |y| ≤ 5 can be solved by creating four separate inequalities: x + y ≤ 5, x - y ≤ 5, -x + y ≤ 5, and -x - y ≤ 5.

Q: Can I use technology to solve inequalities?

A: Yes, you can use technology to solve inequalities. Technology can be used to graph the solution to the inequality on a number line or a coordinate plane, or to solve the inequality using algebraic methods. For example, you can use a graphing calculator to graph the solution to the inequality x ≥ 2, or you can use a computer algebra system to solve the inequality x^2 + 4x + 4 ≤ 0.