Inequalities: Applying the formula

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 $x < 3$.

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

To solve the inequality 6x+d < 3x+1 , we follow these steps:

• Subtract $3x$ from both sides: 3x + d < 1

• Subtract $d$ from both sides: 3x < 1 - d

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

To solve the inequality$7x+c \leq 4x-2$, we follow these steps:

• Subtract $4x$ from both sides: $3x + c \leq -2$

• Subtract $c$ from both sides: $3x \leq -2 - c$

• Divide both sides by $3$: $x \leq -\frac{c+2}{3}$

To solve the given inequalities, we will handle each one individually and then find the valid range of $x$ where both are satisfied:

• First Inequality: $3x + 4 < 9$
  Solve for $x$ by first subtracting 4 from both sides:
  $3x + 4 - 4 < 9 - 4 \Rightarrow 3x < 5$.
  Next, divide both sides by 3 to solve for $x$:
  $x < \frac{5}{3}$.

• Second Inequality: $3 < x + 5$
  Solve for $x$ by subtracting 5 from both sides:
  $3 - 5 < x + 5 - 5 \Rightarrow -2 < x$.

Now we combine the solutions from both inequalities:

• From the first inequality, we have: $x < \frac{5}{3}$.
• From the second inequality, we have: $x > -2$.

Therefore, combining these results, the solution is the intersection of the two ranges:

$-2 < x < \frac{5}{3}$, which can be expressed as $-2 < x < 1\frac{2}{3}$.

The final solution is $-2 < x < 1\frac{2}{3}$.

To solve this problem effectively, we will proceed by solving each inequality separately:

• Step 1: Solve the inequality $-3x + 15 < 3x$.
• Add $3x$ to both sides to get: $15 < 6x$.
• Divide both sides by $6$ to solve for $x$: $x > \frac{15}{6}$, simplifying to $x > 2.5$.
• Step 2: Solve the second inequality $3x < 4x + 8$.
• Subtract $3x$ from both sides to isolate $x$: $0 < x + 8$.
• Subtract $8$ from both sides to find: $-8 < x$.
• Step 3: Find the overlap of the solutions.
• The solution to the inequalities $-8 < x$ and $x > 2.5$ is simply $x > 2.5$ since $x > 2.5$ is more restrictive.

Thus, the values of $x$ that satisfy the original compound inequality are those for which $x > 2.5$.

Therefore, the solution to the problem is $2.5 < x$.

Solving an inequality equation is just like a normal equation. We start by trying to isolate the variable (X).

It is important to note that in this equation there are two variables (X and a), so we may not reach a final result.

 8x+a<3x-4

We move the sections

8x-3x<-4-a

We reduce the terms

5x<-4-a

We divide by 5

x< -a/5 -4/5

And this is the solution!

To solve this problem, we'll break it down into manageable steps:

The problem asks us to find $a$ satisfying two conditions simultaneously: $0 < 8a + 4$ and $8a + 4 \leq -a + 9$.

• Step 1: Solve the first inequality.

The inequality $0 < 8a + 4$ can be simplified by subtracting 4 from both sides:

$-4 < 8a$

Next, divide each side by 8 to isolate $a$:

$-\frac{1}{2} < a$
• Step 2: Solve the second inequality.

The inequality $8a + 4 \leq -a + 9$ can be simplified. Begin by adding $a$ to both sides to gather all $a$-terms on one side:

$9a + 4 \leq 9$

Subtract 4 from both sides:

$9a \leq 5$

Finally, divide each side by 9 to solve for $a$:

$a \leq \frac{5}{9}$
• Step 3: Combine the results of these inequalities.

We now combine the results from step 1 and step 2:

The condition from step 1 is $-\frac{1}{2} < a$.

The condition from step 2 is $a \leq \frac{5}{9}$.

Together, these conditions provide the range:

$-\frac{1}{2} < a \leq \frac{5}{9}$

The solution set is $-\frac{1}{2} < a \leq \frac{5}{9}$.

Therefore, the correct answer choice is: $-\frac{1}{2} < a \leq \frac{5}{9}$.

To solve this problem, we will address each inequality separately and examine the conditions:

Step 1: Solving the First Inequality

The first inequality is $8x < 3x + 9$.

• Subtract $3x$ from both sides to isolate terms with $x$:

$8x - 3x < 9$

$5x < 9$

• Divide both sides by 5 to solve for $x$:

$x < \frac{9}{5}$

Thus, the solution to the first inequality is $x < 1.8$.

Step 2: Solving the Second Inequality

The second inequality is $5x + 4 < 0$.

• Subtract 4 from both sides:

$5x < -4$

• Divide both sides by 5:

$x < -\frac{4}{5}$

Thus, $x < -0.8$.

Step 3: Analyzing the Combined Conditions

• We need to find the values of $x$ that satisfy $x < 1.8$ (from Step 1) but do not satisfy $x < -0.8$ (from Step 2).
• This indicates $x$ should be greater than or equal to $-0.8$ and still less than $1.8$.

Therefore, the solution is $-0.8 \leq x < 1.8$.

Thus, the value of $x$ satisfies the desired condition, which corresponds to choice 3 in the options provided.

Therefore, the solution to the problem is $-0.8 \leq x < 1.8$.