Question Solve the equation $4x - 7x = 5x + 1$ and choose the correct solution number line.

Studdy Solution

STEP 1

Assumptions
1. We are given the equation $4x - 7x = 5x + 1$ .
2. We need to solve for the variable $x$ .
3. After solving the equation, we will also solve the inequality that corresponds to the equation.
4. We will represent the solution on a number line.

STEP 2

Combine like terms on the left side of the equation.
$4x - 7x = 5x + 1$
$-3x = 5x + 1$

STEP 3

Now, we need to isolate the variable $x$ on one side of the equation. To do this, we will subtract $5x$ from both sides of the equation.
$-3x - 5x = 5x + 1 - 5x$

STEP 4

Combine the $x$ -terms on the left side of the equation.
$-8x = 1$

STEP 5

To solve for $x$ , divide both sides of the equation by $-8$ .
$\frac{-8x}{-8} = \frac{1}{-8}$

STEP 6

Calculate the value of $x$ .
$x = -\frac{1}{8}$

STEP 7

Now we have the solution to the equation. Next, we will solve the inequality that corresponds to the original equation. The inequality is:
$4x - 7x < 5x + 1$

STEP 8

Combine like terms on the left side of the inequality, just like we did for the equation.
$-3x < 5x + 1$

STEP 9

Subtract $5x$ from both sides of the inequality to isolate the $x$ -terms on one side.
$-3x - 5x < 5x + 1 - 5x$

STEP 10

Combine the $x$ -terms on the left side of the inequality.
$-8x < 1$

STEP 11

Divide both sides of the inequality by $-8$ , remembering that dividing by a negative number reverses the inequality sign.
$\frac{-8x}{-8} > \frac{1}{-8}$

STEP 12

Calculate the inequality solution.
$x > -\frac{1}{8}$

STEP 13

To represent the solution on a number line, we need to indicate that $x$ is greater than $-\frac{1}{8}$ . We do this by drawing an open circle at $-\frac{1}{8}$ and shading the number line to the right of $-\frac{1}{8}$ to show that $x$ can take any value greater than $-\frac{1}{8}$ .
The solution to the equation is $x = -\frac{1}{8}$ , and the solution to the inequality is $x > -\frac{1}{8}$ . The number line with the correct solution will show an open circle at $-\frac{1}{8}$ and shading to the right.

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