# Solving Inequalities

## Introduction to Solving Inequalities

Inequalities are fundamental components of mathematics and everyday decision-making. They express the relationship between two values, indicating that one is greater than, less than, equal to, or not equal to another. These comparisons form the basis of problem-solving across various disciplines, from personal budgeting to scientific research.

## Understanding Inequalities – Definition

Inequalities are mathematical expressions that compare two values, showing that one value is greater than (>) or less than (<), greater than or equal to (≥), or less than or equal to (≤) another.

Inequality Sign Description
$>$ greater than
$<$ less than
$\geq$ greater than or equal to
$\leq$ less than or equal to
What is the inequality symbol for "greater than or equal to"?
Is the inequality $x > 4$ true if $x = 8$?
For the inequality $y < 10$, is $y = 5$ true or false?
If $z \leq 3$, would $z = 2$ satisfy the inequality?
Given the inequality $a \geq -2$, is $a = -5$ true or false?
Determine whether the inequality $b < 7$ holds true if $b = 10$.

## Solving Linear Inequalities – Examples

Let's solve a linear inequality together:

Given: $3x - 5 ≤ 2x + 10$

Steps:

1. Subtract $2x$ from both sides: $x - 5 ≤ 10$.
2. Add $5$ to both sides: $x ≤ 15$.

Result: $x$ must be less than or equal to $15$.

When solving inequalities, remember to reverse the direction of the inequality sign when multiplying or dividing both sides by a negative number. This ensures that the inequality remains true.

Let's solve another linear inequality:

Given: $-4x + 3 < 15$

Steps:

1. Subtract $3$ from both sides: $-4x < 12$.
2. Divide both sides by $-4$, remembering to reverse the inequality sign due to dividing by a negative number: $\frac{-4x}{-4} > \frac{12}{-4}$.
3. Simplify: $x > -3$.

Result: $x$ must be greater than $-3$.

## Solving Inequalities – Practice

Let’s practice some more!

How would you solve the inequality $2(x - 3) > 4x - 6$?
Solve the inequality $2x - 3 > 5$ for $x$.
Determine the solution set for the inequality $-3y + 7 \leq 10$.
Solve the inequality $\frac{x}{2} + 4 < 6$ for $x$.

## Solving Inequalities – Summary

Key Learnings from this Text:

• Inequalities help us express relationships between two values.
• They come in various forms, including linear, quadratic, absolute value, and more.
• Solving inequalities often mirrors solving equations, with the added rule of flipping the inequality when multiplying or dividing by a negative number.
• Shading on a graph represents the solution set of an inequality.

For more practice with inequalities, check out our interactive exercises and video tutorials that make learning engaging and fun!

## Solving Inequalities – Frequently Asked Questions

What is an inequality in math?
Can you add or subtract the same number from both sides of an inequality?
What happens if you multiply or divide both sides of an inequality by a negative number?
How do you represent the solution of an inequality on a number line?
What's the difference between '<' and '≤' in inequalities?
Are inequalities only used in math?
Can inequalities have more than one solution?
Is solving an inequality the same as solving an equation?
How do you solve a system of inequalities?
Can you have an inequality with no solutions?

## Solving Inequalities exercise

Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the learning text Solving Inequalities.
• ### What does each inequality mean?

Hints

$<$ means less than and $>$ means greater than.

$\leq$ means less than or equal to and $\geq$ means greater than or equal to.

We can write x is greater than 9 as $x > 9$ or $9 < x$.

Solution

$x > 2$ means $x$ is greater than $2$

$x \leq 4$ means $x$ is less than or equal to $4$

$x \geq 4$ means $x$ is greater than or equal to $4$

$x < 2$ means $x$ is less than $2$

$5 > x$ means $x$ is less than $5$

$5 < x$ means $x$ is greater than $5$

• ### True or False?

Hints

To test the inequality, replace the variable (the letter) by the number.

For example, does $a = 5$ satisfy the inequality $a < 7$?

Replace $a$ with $5$, giving $5 < 7$.

$5$ is less than $7$ so this is true.

You can use a number line to help you. Numbers to the left are less than numbers to the right.

In the last example, $5 < 7$ because $5$ is to the left of $7$.

We can also say that, $7 > 5$ because $7$ is to the right of $5$.

Be careful with negative numbers. For example $-4$ is less than $-1$ as $-4$ is to the left of $-1$.

Solution

The inequality $x > 5$, is true when $x$ is $9$?

The inequality $y < 2$, is false when $y = 4$?

When $m > 8$, it is false that $m = 8$ satisfies the inequality?

The inequality $t \geq -5$ is true when $t$ is $-3$?

When $p \leq 7$, it is true that $p$ is $7$ satisfies the equation?

• ### Solve an inequality with a negative variable.

Hints

Solving inequalities is often similar to solving equations.

Remember to reverse the inequality sign when dividing by a negative.

For example: $-5t > 35$

Divide by $-5$ and reverse the inequality giving:

$t < -7$

Solution

$-6p - 3 > 15$

Add $3$ to both sides.

$-6p > 18$

Divide both sides by $-6$, remember to reverse the inequality sign when dividing by a negative.

$p < -3$

Result: p is less than $-3$.

• ### Solve the inequality.

Hints

Solving inequalities is often similar to solving equations.

If the inequality has variables on both sides, we start by addressing this. Remove the smaller variable first by carrying out the inverse operation.

For example: $7y - 6 \geq 2y + 4$.

Step one would be to subtract $2y$ on both sides.

This gives $5y - 6 \geq 4$.

Solution

Solve $5m - 8 \leq 3m + 4$

Step one: Subtract $3m$ on both sides.

$\bf{2m}$ $- 8 \leq 4$

Step two: Add $\bf{8}$ to both sides.

$\bf{2m \leq 12}$

Step three: Divide both sides by $\bf{2}$.

$m \leq$ $\bf{6}$

Result: $m$ is less than or equal to $\bf{6}$.

• ### Solve the inequality.

Hints

Remember to state the result at the end. This means that you write the inequality in words.

For example, $t < 7$ means $t$ is less than $7$.

Solution

$2x + 9 < 15$

Subtract 9 from each side:

$2x$ $< 6$

Divide both sides by 2:

$x$ $< 3$

State the result:

$x$ is less than 3.

• ### Solve an inequality with a bracket.

Hints

It is helpful to expand the bracket first.

$x$ and $1x$ are equivalent.

Remember to use inverse operations to solve the inequality.

Solution

Solve $5x + 10 \geq 4(x + 3)$

$5x + 10 \geq 4x + 12$

$1x + 10 \geq 12$

$x \geq 2$

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