To algebraically express a verbal inequality, we will need to translate the given information into mathematical symbols and operations.

Writing the Inequality

The phrase

at

least

can be expressed with the symbol

\geq,

which will be at the center of our expression.

\dots \geq \dots

On the left-hand side of the inequality symbol, we will translate any verbal expression that comes before the phrase

is

at

least .

If we let

x

represent a number, we can form this expression.

twice a number increased by 4 2 x + 4

On the right-hand side of the inequality symbol, we will translate any verbal expression that comes after

is

at

least .

10 more than the number x + 10

Finally, we can bring these two expressions together to form the inequality.

2 x + 4 \geq x + 10

Solving the Inequality

Using the Properties of Inequality, we will solve the inequality by isolating the variable.

2 x + 4 \geq 10 + x

SubIneq

$LHS - x \geq RHS - x$

x + 4 \geq 10

SubIneq

$LHS - 4 \geq RHS - 4$

x \geq 6

This solution tells us that all values greater than or equal to

6

will satisfy the inequality.

Checking Our Solution

We can check our solution by substituting a few arbitrary values into the inequality translated above. The value satisfies the inequality if the inequality remains true after substituting and simplifying.

$x$	$2 x + 4 \geq 10 + x$	Simplify
$5$	$2 (5) + 4 \geq ? 10 + 5$	$14 ≱ 15$
$6$	$2 (6) + 4 \geq ? 10 + 6$	$16 \geq 16$
$7$	$2 (7) + 4 \geq ? 10 + 7$	$18 \geq 17$

We can conclude that as long as $x$ is greater than or equal to $6,$ the inequality is satisfied.

Exercise

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