Chapter 5.3, Problem 26PPS

To determine

To solve the inequality:

$10\le 4\left(2x+3\right)$

And check the solution.

Answer to Problem 26PPS

  $x\le \left(-\frac{1}{4}\right)$

Explanation of Solution

Given:

$10\le 4\left(2x+3\right)$

Concept Used:

• To check the answer solution for the given inequality
1. First check the solution at end point by converting it into an equation.
2. If the solution is $x<m$ , then pick any value of x less than m and check whether it satisfies the inequality.
3. If the solution is $x>m$ , then pick any value of x greater than m and check whether it satisfies the inequality.

Properties of inequality:

1.$b\le c\Rightarrow ab\le ac,\text{ if }a\ge 0$

2.$b\le c\Rightarrow ab\ge ac,\text{ if }a\le 0$

Calculation:

Let the number be x, twice of a number is denoted as 2x , the sum of the twice of a number and three can be written as:

$2x+3$

Now, then 4 times 2x + 3is written as 4(2x + 3), Then the given statement can be written as:

$10\le 4\left(2x+3\right)$ Or $4\left(2x+3\right)\ge 10$

Divide each side by 4 to solve the value of x:

  $\begin{array}{c}\frac{4\left(2x+3\right)}{4}\le \frac{10}{4}\\ 2x+3\le \frac{5}{2}\\ 2x+3-3\le \frac{5}{2}-3\\ 2x\le -\frac{1}{2}\end{array}$

Again divide the inequality by 2:

  $\begin{array}{l}\frac{2x}{2}\le \frac{1}{2}\left(-\frac{1}{2}\right)\\ x\le \left(-\frac{1}{4}\right)=-0.25\end{array}$

Thus, the inequality is true for xis less than and equals to -0.25, thus the solution of given inequality in interval form is:

$\left(-\infty ,-0.25\right]$

Now, to check the answer solution for the given inequality first check the solution at end point by converting it into an equation. Putting x equals to -0.25 in left hand side:

  $\begin{array}{l}\text{At }x=-0.25:\text{}4\left(2x+3\right)\ge 10\\ 4\left(2\left(-0.25\right)+3\right)=4\left(2.5\right)=10\\ \text{Thus }4\left(\left(-0.5\right)+3\right)=10\end{array}$

Put any value any value of x less than-0.25, say -0.20 and check whether it satisfies the inequality, as

  $\begin{array}{l}\text{At }x=-0.20:\text{}4\left(2x+3\right)\ge 10\\ 4\left(2\left(-0.20\right)+3\right)=4\left(2.6\right)=10\\ \text{Thus }10.4>10\end{array}$

Since, the condition satisfied thus solution of inequality is correct.