Inequalities - Examples, Exercises and Solutions

To solve the inequality $7 + 2x < 15$, we start by isolating the variable $x$.

First, subtract 7 from both sides of the inequality:

$7 + 2x - 7 < 15 - 7$

Simplifying this, we get:

$2x < 8$

Next, divide both sides by 2 to solve for $x$:

$\frac{2x}{2} < \frac{8}{2}$

Thus, the solution is:

$x < 4$

This is an inequality problem. The inequality is actually an exercise we solve in a completely normal way, except in the case that we multiply or divide by negative.

Let's start by moving the sections:

5X+8<9

5X<9-8

5X<1

We divide by 5:

X<1/5

And this is the solution!

Inequality equations will be solved like a regular equation, except for one rule:

If we multiply the entire equation by a negative, we will reverse the inequality sign.

We start by moving the sections, so that one side has the variables and the other does not:

-3x>-10-5

-3x>-15

Divide by 3

-x>-5

Divide by negative 1 (to get rid of the negative) and remember to reverse the sign of the equation.

x<5

To solve the inequality $3x+4 \leq 10$, follow these steps:

1. Subtract 4 from both sides: $3x \leq 6$.

2. Divide both sides by 3: $x \leq 2$.

To solve the inequality $2x-5 > 3$, follow these steps:

1. Add 5 to both sides: $2x > 8$.

2. Divide both sides by 2: $x > 4$.

To solve the inequality $4x+3 \geq 11$, follow these steps:

1. Subtract 3 from both sides: $4x \geq 8$.

2. Divide both sides by 4: $x \geq 2$.

To solve the inequality $6x - 2 < 4$, follow these steps:

1. Add 2 to both sides: $6x < 6$.

2. Divide both sides by 6: $x < 1$.

To solve the inequality $4x + 7 > 19$, follow these steps:

1. Subtract 7 from both sides to isolate the term with $x$ on one side.

$4x + 7 - 7 > 19 - 7$

2. This simplifies to:

$4x > 12$

3. Next, divide each side by 4 to solve for $x$.

$x > \frac{12}{4}$

4. This simplifies further:

$x > 3$

Therefore, the solution is $x > 3$.

To solve the inequality $-2x + 9 \leq 3$, follow these steps:

1. Subtract 9 from both sides:

$-2x + 9 - 9 \leq 3 - 9$

2. This simplifies to:

$-2x \leq -6$

3. Divide each side by -2, remembering to reverse the inequality sign since dividing by a negative number:

$x \geq \frac{-6}{-2}$

4. Simplifying gives:

$x \geq 3$

Thus, the solution is $x \leq 3$.

To solve the inequality $6x - 4 < 14$, follow these steps:

1. Add 4 to both sides:

$6x - 4 + 4 < 14 + 4$

2. Simplify:

$6x < 18$

3. Divide both sides by 6 to solve for $x$:

$x < \frac{18}{6}$

4. This simplifies to:

$x < 3$

Thus, the solution is $x < 3$.

First, we will move the elements:

5-8x>7x+3

5-3>7x+8x
2>15x

We divide the answer by 13, and we get:

x > \frac{2}{15}

The task is to interpret the inequality shown by a number line diagram.

First, observe the number line diagram provided. The numbers -4 and 3 are highlighted with vertical dashed lines. A critical point is at 3, where the circle is filled, indicating the inclusion of this point in the set. The line then extends from 3 to the right, suggesting that any point greater than or equal to 3 is included.

This indicates the inequality for $x$ is $x \geq 3$. The filled circle means 3 itself is part of the solution.

Thus, the solution to the inequality represented by the diagram is:

$3 \leq x$

This matches with choice number 3 in the provided options: $3 ≤ x$.

In the exercise, we have an inequality equation.

We treat the inequality as an equation with the sign -=,

And we only refer to it if we need to multiply or divide by 0.

 $10x-4 ≤ -3x-8$

We start by organizing the sections:

$10x+3x-4 ≤ -8$

$13x-4 ≤ -8$

$13x ≤ -4$

Divide by 13 to isolate the X

$x≤-\frac{4}{13}$

Let's look again at the options we were asked about:

Answer A is with different data and therefore was rejected.

Answer C shows a case where X is greater than$-\frac{4}{13}$, although we know it is small, so it is rejected.

Answer D shows a case (according to the white circle) where X is not equal to$-\frac{4}{13}$, and only smaller than it. We know it must be large and equal, so this answer is rejected.

Therefore, answer B is the correct one!

To solve the problem and determine the inequality represented by the number line, follow these steps:

• Examine the endpoints of the interval on the number line. At $-7$, there is an open circle, indicating that $-7$ is not included in the interval.
• At $0$, there is a closed circle, indicating that $0$ is included in the interval.
• This gives us the inequality for the interval: open at $-7$ ($<$) and closed at $0$ ($\leq$).

Therefore, the inequality represented by the number line is $-7 < x \leq 0$.

This is consistent with option 1 in the provided choices. The inequality is represented by $-7 < x \leq 0$.

To solve the inequality 5x+b > 2x+3 , we will follow these steps:

• Subtract $2x$ from both sides: 3x + b > 3

• Subtract $b$ from both sides: 3x > 3 - b

• Divide both sides by $3$: x > -\frac{1}{3}b + 1